This is a vector:

A vector has magnitude (size) and direction :

The length of the line shows its magnitude and the arrowhead points in the direction.

Play with one here:

We can add two vectors by joining them head-to-tail:

And it doesn't matter which order we add them, we get the same result:

Example: A plane is flying along, pointing North, but there is a wind coming from the North-West.

The two vectors (the velocity caused by the propeller, and the velocity of the wind) result in a slightly slower ground speed heading a little East of North.

If you watched the plane from the ground it would seem to be slipping sideways a little.

Have you ever seen that happen? Maybe you have seen birds struggling against a strong wind that seem to fly sideways. Vectors help explain that.

Velocity , acceleration , force and many other things are vectors.

Subtracting

We can also subtract one vector from another:

  • first we reverse the direction of the vector we want to subtract,
  • then add them as usual:

A vector is often written in bold , like a or b .

Calculations

Now ... how do we do the calculations?

The most common way is to first break up vectors into x and y parts, like this:

The vector a is broken up into the two vectors a x and a y

(We see later how to do this.)

Adding Vectors

We can then add vectors by adding the x parts and adding the y parts :

The vector (8, 13) and the vector (26, 7) add up to the vector (34, 20)

Example: add the vectors a = (8, 13) and b = (26, 7)

c = (8, 13) + (26, 7) = (8+26, 13+7) = (34, 20)

When we break up a vector like that, each part is called a component :

Subtracting Vectors

To subtract, first reverse the vector we want to subtract, then add.

Example: subtract k = (4, 5) from v = (12, 2)

a = v + − k

a = (12, 2) + −(4, 5) = (12, 2) + (−4, −5) = (12−4, 2−5) = (8, −3)

Magnitude of a Vector

The magnitude of a vector is shown by two vertical bars on either side of the vector:

OR it can be written with double vertical bars (so as not to confuse it with absolute value):

We use Pythagoras' theorem to calculate it:

| a | = √( x 2 + y 2 )

Example: what is the magnitude of the vector b = (6, 8) ?

| b | = √( 6 2 + 8 2 ) = √(36+64) = √100 = 10

A vector with magnitude 1 is called a Unit Vector .

Vector vs Scalar

A scalar has magnitude (size) only .

Scalar: just a number (like 7 or −0.32) ... definitely not a vector.

A vector has magnitude and direction , and is often written in bold , so we know it is not a scalar:

  • so c is a vector, it has magnitude and direction
  • but c is just a value, like 3 or 12.4

Example: k b is actually the scalar k times the vector b .

Multiplying a vector by a scalar.

When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big or small the vector is.

Example: multiply the vector m = (7, 3) by the scalar 3

It still points in the same direction, but is 3 times longer

(And now you know why numbers are called "scalars", because they "scale" the vector up or down.)

Multiplying a Vector by a Vector (Dot Product and Cross Product)

How do we multiply two vectors together? There is more than one way!

  • The scalar or Dot Product (the result is a scalar).
  • The vector or Cross Product (the result is a vector).

(Read those pages for more details.)

More Than 2 Dimensions

Vectors also work perfectly well in 3 or more dimensions:

Example: add the vectors a = (3, 7, 4) and b = (2, 9, 11)

c = (3, 7, 4) + (2, 9, 11) = (3+2, 7+9, 4+11) = (5, 16, 15)

Example: what is the magnitude of the vector w = (1, −2, 3) ?

| w | = √( 1 2 + (−2) 2 + 3 2 ) = √(1+4+9) = √14

Here is an example with 4 dimensions (but it is hard to draw!):

Example: subtract (1, 2, 3, 4) from (3, 3, 3, 3)

(3, 3, 3, 3) + −(1, 2, 3, 4) = (3, 3, 3, 3) + (−1,−2,−3,−4) = (3−1, 3−2, 3−3, 3−4) = (2, 1, 0, −1)

Magnitude and Direction

We may know a vector's magnitude and direction, but want its x and y lengths (or vice versa):

You can read how to convert them at Polar and Cartesian Coordinates , but here is a quick summary:

Sam and Alex are pulling a box.

  • Sam pulls with 200 Newtons of force at 60°
  • Alex pulls with 120 Newtons of force at 45° as shown

What is the combined force , and its direction?

Let us add the two vectors head to tail:

First convert from polar to Cartesian (to 2 decimals):

Sam's Vector:

  • x = r × cos( θ ) = 200 × cos(60°) = 200 × 0.5 = 100
  • y = r × sin( θ ) = 200 × sin(60°) = 200 × 0.8660 = 173.21

Alex's Vector:

  • x = r × cos( θ ) = 120 × cos(−45°) = 120 × 0.7071 = 84.85
  • y = r × sin( θ ) = 120 × sin(−45°) = 120 × -0.7071 = −84.85

Now we have:

(100,173.21) + (84.85, −84.85) = (184.85, 88.36)

That answer is valid, but let's convert back to polar as the question was in polar:

  • r = √ ( x 2 + y 2 ) = √ ( 184.85 2 + 88.36 2 ) = 204.88
  • θ = tan -1 ( y / x ) = tan -1 ( 88.36 / 184.85 ) = 25.5°

They might get a better result if they were shoulder-to-shoulder!

  • Varsity Tutors
  • K-5 Subjects
  • Study Skills
  • All AP Subjects
  • AP Calculus
  • AP Chemistry
  • AP Computer Science
  • AP Human Geography
  • AP Macroeconomics
  • AP Microeconomics
  • AP Statistics
  • AP US History
  • AP World History
  • All Business
  • Business Calculus
  • Microsoft Excel
  • Supply Chain Management
  • All Humanities
  • Essay Editing
  • All Languages
  • Mandarin Chinese
  • Portuguese Chinese
  • Sign Language
  • All Learning Differences
  • Learning Disabilities
  • Special Education
  • College Math
  • Common Core Math
  • Elementary School Math
  • High School Math
  • Middle School Math
  • Pre-Calculus
  • Trigonometry
  • All Science
  • Organic Chemistry
  • Physical Chemistry
  • All Engineering
  • Chemical Engineering
  • Civil Engineering
  • Computer Science
  • Electrical Engineering
  • Industrial Engineering
  • Materials Science & Engineering
  • Mechanical Engineering
  • Thermodynamics
  • Biostatistics
  • College Essays
  • High School
  • College & Adult
  • 1-on-1 Private Tutoring
  • Online Tutoring
  • Instant Tutoring
  • Pricing Info
  • All AP Exams
  • ACT Tutoring
  • ACT Reading
  • ACT Science
  • ACT Writing
  • SAT Tutoring
  • SAT Reading
  • SAT Writing
  • GRE Tutoring
  • NCLEX Tutoring
  • Real Estate License
  • And more...
  • StarCourses
  • Beginners Coding
  • Early Childhood
  • For Schools Overview
  • Talk with Our Team
  • Reviews & Testimonials
  • Press & Media Coverage
  • Tutor/Instructor Jobs
  • Corporate Solutions
  • About Nerdy
  • Become a Tutor

Solving Problems with Vectors

We can use vectors to solve many problems involving physical quantities such as velocity, speed, weight, work and so on.

The velocity of moving object is modeled by a vector whose direction is the direction of motion and whose magnitude is the speed.

A ball is thrown with an initial velocity of 70 feet per second., at an angle of 35 ° with the horizontal. Find the vertical and horizontal components of the velocity.

Let v represent the velocity and use the given information to write v in unit vector form:

v   = 70 ( cos ( 35 ° ) ) i + 70 ( sin ( 35 ° ) ) j

Simplify the scalars, we get:

v   ≈ 57.34 i + 40.15 j

Since the scalars are the horizontal and vertical components of v ,

Therefore, the horizontal component is 57.34 feet per second and the vertical component is 40.15 feet per second.

Force is also represented by vector. If several forces are acting on an object, the resultant force experienced by the object is the vector sum of these forces.

Two forces F 1 and F 2 with magnitudes 20 and 30   lb , respectively, act on an object at a point P as shown. Find the resultant forces acting at P .

First we write F 1 and F 2 in component form:

v ≈ 57.34 i + 40.15 j

F 1 = ( 20 cos ( 45 ° ) ) i + ( 20 sin ( 45 ° ) ) j = 20 ( 2 2 ) i + 20 ( 2 2 ) j = 10 2 i + 10 2 j F 2 = ( 30 cos ( 150 ° ) ) i + ( 30 sin ( 150 ° ) ) j = 30 ( − 3 2 ) i + 30 ( 1 2 ) j = − 15 3 i + 15 j

So, the resultant force F is

F = F 1 + F 2 = ( 10 2   i + 10 2 j ) + ( − 15 3   i + 15 j ) = ( 10 2 − 15 3 ) i + ( 10 2 + 15 ) j ≈ − 12 i + 29 j

A force is given by the vector F = ⟨ 2 , 3 ⟩ and moves an object from the point ( 1 , 3 ) to the point ( 5 , 9 ) . Find the work done.

First we find the Displacement.

The displacement vector is

D = ⟨ 5 − 1 , 9 − 3 ⟩ = ⟨ 4 , 6 ⟩ .

By using the formula, the work done is

W = F ⋅ D = ⟨ 2 , 3 ⟩ ⋅ ⟨ 4 , 6 ⟩ = 26

If the unit of force is pounds and the distance is measured in feet, then the work done is 26 ft-lb.

Download our free learning tools apps and test prep books

bing

Cambridge University Faculty of Mathematics

Or search by topic

Number and algebra

  • The Number System and Place Value
  • Calculations and Numerical Methods
  • Fractions, Decimals, Percentages, Ratio and Proportion
  • Properties of Numbers
  • Patterns, Sequences and Structure
  • Algebraic expressions, equations and formulae
  • Coordinates, Functions and Graphs

Geometry and measure

  • Angles, Polygons, and Geometrical Proof
  • 3D Geometry, Shape and Space
  • Measuring and calculating with units
  • Transformations and constructions
  • Pythagoras and Trigonometry
  • Vectors and Matrices

Probability and statistics

  • Handling, Processing and Representing Data
  • Probability

Working mathematically

  • Thinking mathematically
  • Mathematical mindsets
  • Cross-curricular contexts
  • Physical and digital manipulatives

For younger learners

  • Early Years Foundation Stage

Advanced mathematics

  • Decision Mathematics and Combinatorics
  • Advanced Probability and Statistics

Published 2014

Solver Title

Practice

Generating PDF...

  • Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Mean, Median & Mode
  • Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
  • Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
  • Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
  • Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
  • Linear Algebra Matrices Vectors
  • Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
  • Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
  • Physics Mechanics
  • Chemistry Chemical Reactions Chemical Properties
  • Finance Simple Interest Compound Interest Present Value Future Value
  • Economics Point of Diminishing Return
  • Conversions Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time
  • Pre Algebra
  • Pre Calculus
  • Linear Algebra
  • Add, Subtract
  • Multiply, Power
  • Determinant
  • Minors & Cofactors
  • Characteristic Polynomial
  • Gauss Jordan (RREF)
  • Row Echelon
  • LU Decomposition
  • Eigenvalues
  • Eigenvectors
  • Diagonalization
  • Exponential
  • Scalar Multiplication
  • Dot Product
  • Cross Product
  • Scalar Projection
  • Orthogonal Projection
  • Gram-Schmidt
  • Trigonometry
  • Conversions

Click to reveal more operations

Most Used Actions

Number line.

  • \begin{pmatrix}1&0&3\end{pmatrix}+\begin{pmatrix}-1&4&2\end{pmatrix}
  • (-3)\cdot \begin{pmatrix}1&5&0\end{pmatrix}
  • |\begin{pmatrix}2&4&-2\end{pmatrix}|
  • \begin{pmatrix}1&2&3\end{pmatrix}\times\begin{pmatrix}1&5&7\end{pmatrix}
  • angle\:\begin{pmatrix}2&-4&-1\end{pmatrix},\:\begin{pmatrix}0&5&2\end{pmatrix}
  • unit\:\begin{pmatrix}2&-4&1\end{pmatrix}
  • projection\:\begin{pmatrix}1&2\end{pmatrix},\:\begin{pmatrix}3&-8\end{pmatrix}
  • scalar\:projection\:\begin{pmatrix}1&2\end{pmatrix},\:\begin{pmatrix}3&-8\end{pmatrix}
  • What are vectors in math?
  • In math, a vector is an object that has both a magnitude and a direction. Vectors are often represented by directed line segments, with an initial point and a terminal point. The length of the line segment represents the magnitude of the vector, and the arrowhead pointing in a specific direction represents the direction of the vector.
  • What are the types of vectors?
  • The common types of vectors are cartesian vectors, column vectors, row vectors, unit vectors, and position vectors.
  • How do you add two vectors?
  • To add two vectors, add the corresponding components from each vector. Example: the sum of (1,3) and (2,4) is (1+2,3+4), which is (3,7)

vector-calculator

  • Advanced Math Solutions – Vector Calculator, Advanced Vectors In the last blog, we covered some of the simpler vector topics. This week, we will go into some of the heavier... Read More

Please add a message.

Message received. Thanks for the feedback.

One to one maths interventions built for KS4 success

Weekly online one to one GCSE maths revision lessons now available

In order to access this I need to be confident with:

Addition and subtraction

Multiplication and division

This topic is relevant for:

GCSE Maths

Here we will learn about vectors , including what vectors are and how to use vectors to solve geometry problems.  We will also learn how to add, subtract and multiply them.

There are also vector worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are vectors?

Vectors describe a movement from one point to another. We need to be able to identify two characteristics of vectors, magnitude and direction . Magnitude is ‘how large’ something is. 

We can find the magnitude of a vector by finding the length of the line segment. We can identify the direction of a vector from the direction of the arrow on line segment that represents it.

how to solve questions on vector

What is a vector?

What is a vector?

1. Vector notation

Vectors can be represented by a straight line segment with an arrow to show the direction of the vector (a directed line segment).  These are also known as Euclidean vectors.

This diagram shows a vector representing the move from point A to point B.

In maths we use special notation to write the vector A to B.  Boldface is often used in textbooks.

If we reverse the direction of the arrow, so that it is in the opposite direction, we get the negative of vector \textbf{a} .

Step-by-step guide: Vector notation

2. Column vector

We use column vectors to give information about a vector.

A vector has two components: 

There is a horizontal component, also known as the x component.  This is the top number in the column vector and tells us how many spaces to the right or left to move. If the number is positive, the direction is to the right.  If the number is negative, the direction is to the left.

There is a vertical component, also known as the y component.  This is the bottom number in the column vector and tells us how many spaces up or down to move.  If the number is positive, the direction is upwards.  If the number is negative, the direction is downwards.

This tells us to go 5 to the right and 2 up.

This tells us to go 5 to the left and 2 down.

Step-by-step guide: Column vector

3. Magnitude of a vector

We can find the magnitude of a vector (the length of the arrow).  Notice that the vector components make a right-angled triangle.  We can use vector components and Pythagoras’ theorem to find the magnitude of the vector.

The length of a vector is its absolute value and we use the modulus symbol. 

Here is the formula:

If the magnitude is equal to 1, then the vector is known as a unit vector.

If the magnitude is equal to 0 , then the vector is known as a zero vector .

Step-by-step guide: Magnitude of a vector

Vector arithmetic 

4. vector addition.

Here is vector \textbf{a} and vector \textbf{b} .

We can add the vectors :

The vector \textbf{a} + \textbf{b} is known as the resultant vector.

Step-by-step guide: Vector addition

5. Vector subtraction

We can subtract vectors . 

When subtracting vectors, the order is important.

Step-by-step guide: Vector subtraction

6. Vector multiplication

We can multiply vectors by a scalar . 

Step-by-step guide: Vector multiplication

7. Combining vector addition, subtraction and multiplication

Vector addition, subtraction and multiplication are often combined.

A vector quantity you may meet in GCSE Science is velocity vector.  

A scalar quantity would be speed.

Step-by-step guide: Vector problems

8. Vector geometry

We can solve geometrical problems using vectors. Vectors are equal if they have the same magnitude and direction regardless of where they are.

how to solve questions on vector

Vectors A and B are equal. They are travelling in the same direction and have the same magnitude (length).

OBDE is a parallelogram. A is the midpoint of OE and C is the midpoint of BD .

OAFG is a parallelogram. C is the midpoint of AF and B is the midpoint of OG .

how to solve questions on vector

Vector \overrightarrow{ED} , \overrightarrow{AC} and \overrightarrow{OB} are equal. They are parallel, so they travel in the same directions . They also have the same magnitude . 

Vectors \overrightarrow{OA} , \overrightarrow{BC} , \overrightarrow{AE} , \overrightarrow{CD} and \overrightarrow{GF} are also equal.

how to solve questions on vector

Vectors b and c are not equal. They do not have the same magnitude (length) and they do not travel in the same direction .

Negative vectors have the same magnitude but travel in the opposite direction. 

how to solve questions on vector

The vectors have the same magnitude but are travelling in the opposite direction, denoted by the arrows.

How to use vectors

In order to use vectors consider:

  • If there is a diagram, read the information, add other vectors, check the route and simplify your answer if needed.
  • Remember column vectors have two vector components.
  • If you are asked to find the magnitude of a vector use Pythagoras’ theorem.
  • If the question requires addition, subtracting and multiplying vectors – take care with the order of operations.

How to use vectors

Vectors worksheet

Get your free vectors worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Vectors examples

Example 1: vector notation.

Here is a parallelogram. 

Write the vector \overrightarrow{CO} in terms of \textbf{a} and \textbf{b} .

  • Use the information that the shape is a parallelogram to add in more vectors.
  • Check the route – we need to start at point C and go to point O along the vectors.

Example 2: column vector

Write vector \textbf{v} as a column vector

  • Vector \textbf{v} has two components, a horizontal component ( x component) and a vertical component ( y component). 
  • From the starting point at the bottom right point draw a horizontal line and a vertical line to make a right-angled triangle.
  • We write the horizontal component first, here we have moved 3 to the left so we write this as -3 .
  • Next we write the vertical component, here we have moved 4 upwards so we write this as 4 .

Example 3: magnitude of a vector

Calculate the magnitude of vector \textbf{c} , give your answer to 2 decimal places.

  • We need to use Pythagoras’ theorem.

So the magnitude of vector \textbf{c} is 5.83 (to 2dp )

Example 4: vector addition

Here are two vectors, \textbf{a} and \textbf{b}

Work out \textbf{a}+\textbf{b}

  • Add the x components and add the y components.

Example 5: vector subtraction

Here are two vectors \textbf{v} and \textbf{w}

Work out \textbf{w} - \textbf{v}

  • Subtract the vector components. Make sure you get the vector subtraction in the correct order.

Example 6: vector multiplication

Here is the vector \textbf{f}

Work out 4 f

  • Multiply both the vector components by the scalar (number)

Example 7: combining vector addition, subtraction and multiplication

Here are two vectors \textbf{c} and \textbf{d}

Work out 2\textbf{c}-4\textbf{d}

  • Work out 2\textbf{c} and 4\textbf{d} separately and then subtract them.
  • So, 2\textbf{c}-4\textbf{d} would be
  • Or alternatively you can work them out in one line.

Common misconceptions

  • Column vectors notation

2D column vectors only have 2 numbers within the brackets. Column vectors have the top number and the bottom number in the brackets.

There is no need for any other punctuation marks such as commas or semicolons.  There is no need for a line to separate the numbers.

  • Vector addition order

Vector addition is commutative.  This means that the order in which we add vectors is not important. 

  • Order of the subtraction

The order in which you subtract vectors is very important. It is NOT like vector addition where the order is unimportant.  Vector subtraction is NOT commutative.

  • A component of a vector can be zero

Vector components can be zero.

If both components of a vector are zero, this vector is known as the zero vector.

  • Direction of the vector

Check the arrow on vector diagrams to see the direction of the vector.  Check for negative signs in column vectors.

  • Be careful calculating with negative numbers

It is easy to make errors with negative numbers when adding, subtracting, multiplying or squaring.  Brackets around the negative number can make calculations more accurate. 

Practice vectors questions

1. Write the vector \overrightarrow{BC} in terms of \textbf{a} and \textbf{b}:

GCSE Quiz False

Go from point B to point C, via points A and D

2. Write this vector as a column vector:

We are aiming to draw a right-angled triangle with the starting point and the end point of the vector. Draw a horizontal line from the starting point. Because the line goes 2 squares to the left the top number of the column vector will be -2.

Then draw a vertical line to the end point. Because the line goes 3 squares upwards the bottom number of the column vector will be 3 .

3. What is the magnitude of the following vector? Give your answer to 3 significant figures.

The magnitude is calculated by using Pythagoras’ theorem.

4. Work out:

Add the x components and add the y components

5. Work out:

6. Work out:

Each of the components are multiplied by the scalar number in front of the column vector.

Vectors GCSE questions

1. The vector \textbf{c} is drawn on the grid.

(a) Write vector \textbf{c} as a column vector.

(b) From point P, draw the vector 3 \textbf{c} .

2. Here is trapezium

(a) Find, in terms of \textbf{a} , the vector \overrightarrow{AO} .

(b) Find, in terms of \textbf{a} and \textbf{b} , the vector \overrightarrow{OC} .

(c) Find, in terms of \textbf{a} and \textbf{b} , the vector \overrightarrow{AC} . Give your answer in its simplest form.

3. Here are two vectors

(a) Write down as a column vector, \textbf{a}+\textbf{b}

(b) Write down as a column vector, 3\textbf{a}+2\textbf{b}  

Learning checklist

You have now learned how to:

  • Write about vectors using vector notation
  • Write vectors as column vectors
  • Calculate the magnitude of a vector using Pythagoras’ theorem
  • Add vectors
  • Subtract vectors
  • Multiply a vector by a scalar
  • Solve geometry problems using vectors

The next lessons are

  • Loci and construction
  • Transformations
  • Circle theorems

Did you know?

Vectors are very useful and can be extended beyond GCSE mathematics. Vector analysis is the branch of mathematics that studies vectors. 

At GCSE we study two-dimensional vectors, but we can also look at three-dimensional vectors.

In A Level maths cartesian coordinates are also referred to as position vectors when we use a coordinate system as our vector space. In maths a vector is an element of a vector space.

Vectors can also be extended further by learning how to multiply two vectors together using the dot product.  This is also known as the scalar product of two vectors.  It is possible to multiply vectors another way is known as a cross product.  This is also known as the vector product of two vectors.

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

GCSE Benefits

Find out more about our GCSE maths tuition programme.

Privacy Overview

logo white

  • Mathematicians
  • Math Lessons
  • Square Roots
  • Math Calculators
  • Vector equations – Explanation and Examples

JUMP TO TOPIC

What Is A Vector Equation?

Vector equation of a straight line, practice problems, vector equations – explanation and examples.

Vector Equations

“The vector equation is an equation of vectors which is when solved, gives the result in the form of a vector.”

In this topic, we shall briefly discuss the following mentioned concepts:

  • What is a vector equation?
  • How to solve a vector equation?
  • What is a vector equation of a straight line?
  • What is a vector equation of a circle?

A vector equation is an equation involving n numbers of vectors. More formally, it can be defined as an equation involving a linear combination of vectors with possibly unknown coefficients, and upon solving, it gives a vector in return.

Generally, a vector equation is defined as “Any function that takes anyone or more variable and in return gives a vector.” 

Any vector equation involving vectors with n number of coordinates is similar to the linear equation system with n number of coordinates involving numbers. For example,

Consider a vector equation,

r <4,5,6> + t<3,4,1> = <8,5,9>

It can also be written as

<4r,5r,6r> + <3t,4t,1t> =<8,5,9>

<4r+3t, 5r+4t, 6r+1t> = <8,5,9>

For two vectors to be equal, all the coordinates must be equal, so it can also be written as a system of linear equations. Such a representation is as follows:

So, the vector equation can be solved by converting it into a system of linear equations. Hence, it simplifies and becomes easier to solve. 

In our daily life, vectors play a vital role. Most of the physical quantities used are vector quantities. Vectors have many true applications, including the situations designated by force and velocity. For example, if a car is moving on a road, various forces will be acting on it. Some forces act in the forward direction and some in the backward direction to balance the system. So, all these forces are vector quantities. We use vector equations to find out various physical quantities in 2-D or 3-D, such as velocity, acceleration, momentum, etc.

Vector equations give us a diverse and more geometric way of viewing and solving the linear system of equations.

Overall, we can conclude that the vector equation is:

x 1. t 1 +x 2 .t 2 +···+x k .t k = b

where t 1 ,t 2 ,…,t k ,b are vectors in Rn and x 1 ,x 2 ,…,x k are unknown scalars, has the same solution set as the linear system with an augmented matrix of the given equation .

Therefore, the vector equation is given as,

r = r 0 +k v

Let’s understand this concept with the help of examples.

A car moves with a constant velocity on a straight road initially at time t=2 the position vector of the car is (1,3,5) then after some time at t=4, the car’s position vector is described as (5,6,8). Write down the vector equation of the position of the object. Also, express it in the form of parametric equations.

Since the vector equation of a straight-line is given as 

r = r 0 +t v

r 0 = <1,3,5>

r = <5,6,8>

<5,6,8> = <1,3,5> + 4 v

<5,6,8> – <1,3,5> = 4 v

<4,3,3> = 4 v

v = <1,3/4,3/4>

Now, finding vector equation of object’s position

r = <1,3,5> + t<1,3/4,3/4>

where vector r is <x,y,z>

<x,y,z> = <1,3,5> + <1t,3/4t,3/4t>

Expressing in the form of the parametric equation:

As two vectors are only equivalent if their coordinates are equal. So, due to equality, we can write as,

The vector equation of lines identifies the position vector of line with reference to the origin and direction vector and we can find out the dimensions of vectors corresponding to any length. This works for the straight lines and curves.

Note: The position vector is used to describe the position of the vector. It is a straight line having one end fixed and the other attached to the moving vector to specify its position.

Write down the following equations as vector equations

Let’s consider equation 1 first:

Since the equation given above is an equation of a straight-line:

Firstly, we will select two points on the given line.

Let’s simplify the equation,

So, the first point is s (7,0) or OS (7,0)

Now let find out the second point that is halfway through the first point then,

14 = -2y + 7

So, the second point T (14, -3.5) or OT (14, -3.5)

OS – OT = (7,0) – (14, -3.5)

OS – OT = (-7, 3.5)

So, the vector equation form of the above equation is,

R = <7,0> + k<-7,3.5>

    R = <7-7k, 3.5k>

Now, let’s solve equation 2:

Since the equation given above is an equation of a straight-line

So, the first point is s (2,0) or OS (2,0)

So, the second point T (4, -5/2) or OT (4, -5/2)

OS – OT = (2,0) – (4, -5/2)

OS – OT = (-2, 5/2)

R = <2,0> + k<-2,5/2>

  R = <2-2k, 5/2k>

Now, let’s do equation 3:

x = -3/5y+8

So, the first point is s (8,0) or OS (8,0)

16 = -3/5y+8

-3/5y = 16-8

So, the second point T (16, -13.33) or OT (16, -13.33)

OS – OT = (8,0) – (16, -13.33)

OS – OT = (-8, 13.33)

R = <8,0> + k<-8,13.33>

  R = <8-8k, 13.33k>

We all are familiar with the equation of the line that is y=mx+c, generally called a slope-intercept form where m is the slope of the line and x and y are the point coordinates or intercepts defined on the x and y axes. However, this form of the equation is not enough to completely explain the line’s geometrical features. That’s why we use a vector equation to describe the position and direction of the line completely. 

To find the points on the line, we will use the method of vector addition. We need to find out the position vector and the direction vector. For the position vector, we will add the position vector of the known point on the line to the vector v that lies on the line, as shown in the figure below.

vector equation of a straight line

So, the position vector r for any point is given as r = op + v

Then, the vector equation is given as 

R = op + k v

Where k is a scalar quantity that belongs from R N , op is the position vector with respect to the origin O, and v is the direction vector. Basically, k tells you how many times you will go the distance from p to q in the specified direction. It can be ½ if half of the distance would be covered and so on.

If two points on the line are known, we can find out the line’s vector equation. Similarly, if we know the position vectors of two points op and oq on a line, we can also determine the vector equation of the line by using the vector subtraction method. 

v = op – oq

Therefore, the equation of vector is given as,

R = op +k v

Let’s solve some examples to comprehend this concept. 

Write down the vector equation of a line through points P (2,4,3) and Q (5, -2,6).

Let the position vector of the given points P and Q with respect to the origin is given as OP and OQ, respectively.

OP = (2,4,3) – (0,0,0)

OP = (2,4,3)

OQ = (5, -2,6) – (0,0,0)

  OQ = (5, -2 ,6)

Since we know that the vector equation of a line is defined as,

R = OP + k v

Where v = OQ – OP

v = (5, -2,6) – (2,4,3)

  v = (3, -6, 3)

So, the vector equation of the straight line is given as,

  R = <2,4,3> + k<3, -6,3>

Determine the vector equation of the line where k=0.75. If the points given on the line are defined as A (1,7) and B (8,6).

vector equation example 4

Let the position vector of the given points A and B with respect to the origin are OA and OB, respectively.

 OA = (1,7) – (0,0)

 OB = (8,6) – (0,0)

  OB = (8,6)

 R = OA +k v

Where v = OB – OA

v = (8,6) – (1,7)

  v = (7, -1)

Where k=0.75

  R = <1,7> + 0.75<7, -1>

Write down the vector equation of a line through points P (-8,5) and Q (9,3).

OP = (-8,5) – (0,0)

OP = (-8,5)

OQ = (9,3) – (0,0)

 R = OP + k v

v = (9,3) – (-8,5)

v = (17, -2)

R = <-8,5> + k<17, -2>

Vector Equation Of A Circle

Earlier, we have discussed the vector equation of a straight line. Now we will discuss the vector equation of a circle having radius r and with some center c, which we generally say that the circle is centered at c (0,0), but it may be located at any other point in the plane.

The vector equation of a circle is given as

r (t) = <x(t), y(t)>

where x(t) = r.cos(t) and y(t) = r.sin(t), r is the radius of the circle and t is the defined as the angle.

Let us consider a circle with center c and radius r, as shown in the figure below.

vector equation of a circle

The position vector of the radius and center c is given as r and c, respectively. Then the radius of the circle is represented by vector CR, where CR is given as r – c.

Since the radius is given as r so magnitude if CR can be written as 

| CR | = r^ 2

 ( r – c ). ( r – c ) = r^ 2

| r – c | = r

This can also be called a vector equation of a circle.

Write down the vector equation and the cartesian equation of a circle with center c at (5,7) and radius 5m.

Vector equation of a circle:

| r – <5,7>| = 5

( r – <5,7>)^ 2 = 25

Cartesian equation of a circle:

(x-h)^ 2 +(y-k)^ 2 = r 2

(x-5)^ 2 + (y-7)^ 2 = 25

Determine if the point (2,5) lies on the circle with the vector equation of a circle given as | r -<-6,2>| = 3.

We must find out whether the given point lies inside the circle or not provided the circle’s vector equation.

Since putting the value of the point in the given vector equation

= |<2,5>-<-6,2>|

= |<2+6,5-2>|

= |<8,3>|

= √ ((8)^ 2 +(3)^ 2 )

= √ (73) ≠ 3

Hence, the point does not lies inside the circle.

  • Write down the following equations as vector equations :  x=3y+5        x=-9/5y+3        x+9y=4
  • Determine the equation for the line defined by points A (3,4,5) and B (8,6,7). Find the position vector for a point, half-way between the two points.
  • Write a vector equation of the line parallel to vector Q and passing through point o with the given position vector P .

Q = <-2,6> P = <3, -1> 

Q = <1,8> P = <9, -3>

  • Write down the vector equation of a line through points P (-8/3,5) and Q (5,10).
  • A car moves with a constant velocity on a straight road initially at time t=2 the position vector of the car is (1/2,8) then after some time at t=4, the car’s position vector is described as (5,10). Write down the vector equation of the position of the object. Also, express it in the form of parametric equations.
  • Write down the vector equation and the cartesian equation of a circle with center c at (8,0) and radius 7m.
  • Determine if the point (3,-5) lies on the circle with the vector equation of a circle given as | r -<-3,4>| = 4.
  • (i) . r = <5 – 5k , (-5/3)k            (ii) . r = <3 – 3k, (15/9)k >          (iii) . r = <4 – 4k, (4/9)k >
  • r = <11/2 , 5, 6 >
  • (i) . r = <3, -1> + t<-2, 6>          (ii) . r = <9, -3> + t<1, 8>
  • R = <-8/3, 5> + k<23/3, 5>
  • r = <5, 10> +t <-9/8, -1/2> and x = 5 – (9/8)t , y = 10 – (1/2)t
  • |r – <8, 0>| = 7 and (x – 8) 2 + y 2 =49

All the vector diagrams are constructed by using GeoGebra.

Previous Lesson  |  Main Page | Next Lesson

Math Topics

Numbers and Quantities

Statistics and Probability

How to Solve Vector Equations

To solve vector equations, for each vector, gather the factors that are in front of it. Do this with all the vectors in the equation to make a system of equations .

Vector Equations

A vector equation is expressed as

where u → and v → are two non-parallel vectors, and a , b , c and d are expressions that can include both constants and variables.

Set the expressions in front of u → equal to each other and the expressions in front of v → equal to each other:

This is your system of equations.

Find the values of k and l given that

You put the expressions in front of u → equal to each other and the expressions in front of v → equal to each other. Then you can solve the set of equations:

You put the expressions in front of u → equal to each other and the expressions in front of v → equal to each other. That gives you this system of equations, which you can solve:

White arrow pointing to the Left

Library homepage

  • school Campus Bookshelves
  • menu_book Bookshelves
  • perm_media Learning Objects
  • login Login
  • how_to_reg Request Instructor Account
  • hub Instructor Commons
  • Download Page (PDF)
  • Download Full Book (PDF)
  • Periodic Table
  • Physics Constants
  • Scientific Calculator
  • Reference & Cite
  • Tools expand_more
  • Readability

selected template will load here

This action is not available.

Mathematics LibreTexts

2.3: The span of a set of vectors

  • Last updated
  • Save as PDF
  • Page ID 82482

  • David Austin
  • Grand Valley State University

Our work in this chapter enables us to rewrite a linear system in the form \(A\mathbf x = \mathbf b\text{.}\) Besides being a more compact way of expressing a linear system, this form allows us to think about linear systems geometrically since matrix multiplication is defined in terms of linear combinations of vectors.

We now return, in this and the next section, to the two fundamental questions asked in Question 1.4.2.

  • Existence: Is there a solution to the equation \(A\mathbf x=\mathbf b\text{?}\)
  • Uniqueness: If there is a solution to the equation \(A\mathbf x=\mathbf b\text{,}\) is it unique?

In this section, we focus on the existence question and introduce the concept of span to provide a framework for thinking about it geometrically.

Preview Activity 2.3.1. The existence of solutions.

  • If the equation \(A\mathbf x = \mathbf b\) is inconsistent, what can we say about the pivots of the augmented matrix \(\left[\begin{array}{r|r} A & \mathbf b \end{array}\right]\text{?}\)

If \(\mathbf b=\threevec{2}{2}{5}\text{,}\) is the equation \(A\mathbf x = \mathbf b\) consistent? If so, find a solution.

  • Identify the pivot positions of \(A\text{.}\)
  • For our two choices of the vector \(\mathbf b\text{,}\) one equation \(A\mathbf x = \mathbf b\) has a solution and the other does not. What feature of the pivot positions of the matrix \(A\) tells us to expect this?

The span of a set of vectors

In the preview activity, we considered a \(3\times3\) matrix \(A\) and found that the equation \(A\mathbf x = \mathbf b\) has a solution for some vectors \(\mathbf b\) in \(\mathbb R^3\) and has no solution for others. We will introduce a concept called span that describes the vectors \(\mathbf b\) for which there is a solution.

Since we would like to think about this concept geometrically, we will consider an \(m\times n\) matrix \(A\) as being composed of \(n\) vectors in \(\mathbb R^m\text{;}\) that is,

Remember that Proposition 2.2.4 says that the equation \(A\mathbf x = \mathbf b\) is consistent if and only if we can express \(\mathbf b\) as a linear combination of \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\text{.}\)

Definition 2.3.1

The span of a set of vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) is the set of all linear combinations of the vectors.

In other words, the span of \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) consists of all the vectors \(\mathbf b\) for which the equation

is consistent.

The span of a set of vectors has an appealing geometric interpretation. Remember that we may think of a linear combination as a recipe for walking in \(\mathbb R^m\text{.}\) We first move a prescribed amount in the direction of \(\mathbf v_1\text{,}\) then a prescribed amount in the direction of \(\mathbf v_2\text{,}\) and so on. As the following activity will show, the span consists of all the places we can walk to.

Activity 2.3.2.

Let's look at two examples to develop some intuition for the concept of span.

  • \(a = 2\) and \(b=0\text{?}\)
  • \(a = 1\) and \(b=1\text{?}\)
  • \(a = 0\) and \(b=-1\text{?}\)
  • Can the vector \(\twovec{2}{4}\) be expressed as a linear combination of \(\mathbf v\) and \(\mathbf w\text{?}\) Is the vector \(\twovec{2}{4}\) in the span of \(\mathbf v\) and \(\mathbf w\text{?}\)
  • Can the vector \(\twovec{3}{0}\) be expressed as a linear combination of \(\mathbf v\) and \(\mathbf w\text{?}\) Is the vector \(\twovec{3}{0}\) in the span of \(\mathbf v\) and \(\mathbf w\text{?}\)
  • Describe the set of vectors in the span of \(\mathbf v\) and \(\mathbf w\text{.}\)

have a solution?

  • Can the vector \(\twovec{-2}{2}\) be expressed as a linear combination of \(\mathbf v\) and \(\mathbf w\text{?}\) Is the vector \(\twovec{-2}{2}\) in the span of \(\mathbf v\) and \(\mathbf w\text{?}\)

Let's consider the first example in the previous activity. Here, the vectors \(\mathbf v\) and \(\mathbf w\) are scalar multiples of one another, which means that they lie on the same line. When we form linear combinations, we are allowed to walk only in the direction of \(\mathbf v\) and \(\mathbf w\text{,}\) which means we are constrained to stay on this same line. Therefore, the span of \(\mathbf v\) and \(\mathbf w\) consists only of this line.

With this choice of vectors \(\mathbf v\) and \(\mathbf w\text{,}\) all linear combinations lie on the line shown. This line, therefore, is the span of the vectors \(\mathbf v\) and \(\mathbf w\text{.}\)

We may see this algebraically since the vector \(\mathbf w = -2\mathbf v\text{.}\) Consequently, when we form a linear combination of \(\mathbf v\) and \(\mathbf w\text{,}\) we see that

Therefore, any linear combination of \(\mathbf v\) and \(\mathbf w\) reduces to a scalar multiple of \(\mathbf v\text{,}\) and we have seen that the scalar multiples of a nonzero vector form a line.

In the second example, however, the vectors are not scalar multiples of one another, and we see that we can construct any vector in \(\mathbb R^2\) as a linear combination of \(\mathbf v\) and \(\mathbf w\text{.}\)

With this choice of vectors \(\mathbf v\) and \(\mathbf w\text{,}\) we are able to form any vector in \(\mathbb R^2\) as a linear combination. Therefore, the span of the vectors \(\mathbf v\) and \(\mathbf w\) is the entire plane, \(\mathbb R^2\text{.}\)

Once again, we can see this algebraically. Asking if the vector \(\mathbf b\) is in the span of \(\mathbf v\) and \(\mathbf w\) is the same as asking if the linear system

The augmented matrix for this system is

Since it is impossible to obtain a pivot in the rightmost column, we know that this system is consistent no matter what the vector \(\mathbf b\) is. Therefore, every vector \(\mathbf b\) in \(\mathbb R^2\) is in the span of \(\mathbf v\) and \(\mathbf w\text{.}\)

In this case, notice that the reduced row echelon form of the matrix

has a pivot in every row. When this happens, it is not possible for any augmented matrix to have a pivot in the rightmost column. Therefore, the linear system is consistent for every vector \(\mathbf b\text{,}\) which implies that the span of \(\mathbf v\) and \(\mathbf w\) is \(\mathbb R^2\text{.}\)

Notation 2.3.4.

We will denote the span of the set of vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) by \(\laspan{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\text{.}\)

Pivot positions and span

In the previous activity, we saw two examples, both of which considered two vectors \(\mathbf v\) and \(\mathbf w\) in \(\mathbb R^2\text{.}\) In one example, the \(\laspan{\mathbf v,\mathbf w}\) consisted of a line; in the other, the \(\laspan{\mathbf v,\mathbf w}=\mathbb R^2\text{.}\) We would like to be able to distinguish these two situations in a more algebraic fashion. After all, we will need to be able to deal with vectors in many more dimensions where we will not be able to draw pictures.

The key is found by looking at the pivot positions of the matrix \(\left[\begin{array}{rrrr} \mathbf v_1& \mathbf v_2& \ldots\mathbf v_n \end{array}\right] \text{.}\) In the first example, the matrix whose columns are \(\mathbf v\) and \(\mathbf w\) is

which has exactly one pivot position. We found the \(\laspan{\mathbf v,\mathbf w}\) to be a line, in this case.

In the second example, this matrix is

which has two pivot positions. Here, we found \(\laspan{\mathbf v,\mathbf w}=\mathbb R^2\text{.}\)

These examples point to the fact that the size of the span is related to the number of pivot positions. We will develop this idea more fully in Section 2.4 and Section 3.5 . For now, however, we will examine the possibilities in \(\mathbb R^3\text{.}\)

Activity 2.3.3.

In this activity, we will look at the span of sets of vectors in \(\mathbb R^3\text{.}\)

Sketch the vectors below. Then give a written description of \(\laspan{\mathbf e_1,\mathbf e_2}\) and a rough sketch of it below.

Let's now look at this algebraically by writing write \(\mathbf b = \threevec{b_1}{b_2}{b_3}\text{.}\) Determine the conditions on \(b_1\text{,}\) \(b_2\text{,}\) and \(b_3\) so that \(\mathbf b\) is in \(\laspan{\mathbf e_1,\mathbf e_2}\) by considering the linear system

Explain how this relates to your sketch of \(\laspan{\mathbf e_1,\mathbf e_2}\text{.}\)

  • Give a written description of \(\laspan{\mathbf v_1,\mathbf v_2}\text{.}\)

Form the matrix \(\left[\begin{array}{rrrr} \mathbf v_1 & \mathbf v_2 & \mathbf v_3 \end{array}\right]\) and find its reduced row echelon form.

  • If a set of vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) spans \(\mathbb R^3\text{,}\) what can you say about the pivots of the matrix \(\left[\begin{array}{rrrr} \mathbf v_1& \mathbf v_2& \ldots& \mathbf v_n \end{array}\right]\text{?}\)
  • What is the smallest number of vectors such that \(\laspan{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n} = \mathbb R^3\text{?}\)

This activity shows us the types of sets that can appear as the span of a set of vectors in \(\mathbb R^3\text{.}\)

we must obtain vectors of the form

Since the third component is zero, these vectors form the plane \(z=0\text{.}\)

will form a plane.

has two pivot positions.

has three pivot positions, one in every row. This is significant because it means that if we consider an augmented matrix

there cannot be a pivot position in the rightmost column. This linear system is consistent for every vector \(\mathbf b\text{,}\) which tells us that \(\laspan{\mathbf v_1,\mathbf v_2,\mathbf v_3} = \mathbb R^3\text{.}\)

To summarize, we looked at the pivot positions in the matrix whose columns were the vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\text{.}\) We found that with

  • one pivot position, the span was a line.
  • two pivot positions, the span was a plane.
  • three pivot positions, the span was \(\mathbb R^3\text{.}\)

Once again, we will develop these ideas more fully in the next and subsequent sections. However, we saw that, when considering vectors in \(\mathbb R^3\text{,}\) a pivot position in every row implied that the span of the vectors is \(\mathbb R^3\text{.}\) The same reasoning applies more generally.

Proposition 2.3.5.

Suppose we have vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) in \(\mathbb R^m\text{.}\) Then \(\laspan{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}=\mathbb R^m\) if and only if the matrix \(\left[\begin{array}{rrrr} \mathbf v_1& \mathbf v_2& \ldots& \mathbf v_n \end{array}\right]\) has a pivot position in every row.

This tells us something important about the number of vectors needed to span \(\mathbb R^m\text{.}\) Suppose we have \(n\) vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) that span \(\mathbb R^m\text{.}\) The proposition tells us that the matrix \(A = \left[\begin{array}{rrrr} \mathbf v_1& \mathbf v_2\ldots\mathbf v_n \end{array}\right]\) has a pivot position in every row, such as in this reduced row echelon matrix.

Since a matrix can have at most one pivot position in a column, there must be at least as many columns as there are rows, which implies that \(n\geq m\text{.}\)

For instance, if we have a set of vectors that span \(\mathbb R^{632}\text{,}\) there must be at least 632 vectors in the set.

Proposition 2.3.6.

If a set of vectors span \(\mathbb R^m\text{,}\) there must be at least \(m\) vectors in the set.

This makes sense intuitively. We have thought about a linear combination of a set of vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) as the result of walking a certain distance in the direction of \(\mathbf v_1\text{,}\) followed by walking a certain distance in the direction of \(\mathbf v_2\text{,}\) and so on. If \(\laspan{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n} = \mathbb R^m\text{,}\) this means that we can walk to any point in \(\mathbb R^m\) using the directions \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\text{.}\) It makes sense that we would need at least \(m\) directions to give us the flexibilty needed to reach any point in \(\mathbb R^m\text{.}\)

We defined the span of a set of vectors and developed some intuition for this concept through a series of examples.

  • The span of a set of vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) is the set of linear combinations of the vectors. We denote the span by \(\laspan{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\text{.}\)

has a pivot in every row, then the span of these vectors is \(\mathbb R^m\text{;}\) that is, \(\laspan{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n} = \mathbb R^m\text{.}\)

  • Any set of vectors that spans \(\mathbb R^m\) must have at least \(m\) vectors.

Exercises 2.3.4Exercises

In this exercise, we will consider the span of some sets of two- and three-dimensional vectors.

  • Is \(\mathbf b = \twovec{2}{1}\) in \(\laspan{\mathbf v_1,\mathbf v_2}\text{?}\)
  • Is the vector \(\mathbf b=\threevec{-10}{-1}{5}\) in \(\laspan{\mathbf v_1,\mathbf v_2,\mathbf v_3}\text{?}\)
  • Is the vector \(\mathbf v_3\) in \(\laspan{\mathbf v_1,\mathbf v_2,\mathbf v_3}\text{?}\)
  • Is the vector \(\mathbf b=\threevec{3}{3}{-1}\) in \(\laspan{\mathbf v_1,\mathbf v_2,\mathbf v_3}\text{?}\)
  • Give a written description of \(\laspan{\mathbf v_1,\mathbf v_2,\mathbf v_3}\text{.}\)

Provide a justification for your response to the following questions.

  • Suppose you have a set of vectors \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\text{.}\) Can you guarantee that \(\zerovec\) is in \(\laspan{\mathbf v_1\,\mathbf v_2,\ldots,\mathbf v_n}\text{?}\)
  • Suppose that \(A\) is an \(m \times n\) matrix. Can you guarantee that the equation \(A\mathbf x = \zerovec\) is consistent?
  • What is \(\laspan{\zerovec,\zerovec,\ldots,\zerovec}\text{?}\)

For both parts of this exericse, give a written description of sets of the vectors \(\mathbf b\) and include a sketch.

consistent?

Consider the following matrices:

Do the columns of \(A\) span \(\mathbb R^4\text{?}\) Do the columns of \(B\) span \(\mathbb R^4\text{?}\)

Determine whether the following statements are true or false and provide a justification for your response. Throughout, we will assume that the matrix \(A\) has columns \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\text{;}\) that is,

  • If the equation \(A\mathbf x = \mathbf b\) is consistent, then \(\mathbf b\) is in \(\laspan{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\text{.}\)
  • The equation \(A\mathbf x = \mathbf v_1\) is always consistent.
  • If \(\mathbf v_1\text{,}\) \(\mathbf v_2\text{,}\) \(\mathbf v_3\text{,}\) and \(\mathbf v_4\) are vectors in \(\mathbb R^3\text{,}\) then their span is \(\mathbb R^3\text{.}\)
  • If \(\mathbf b\) can be expressed as a linear combination of \(\mathbf v_1, \mathbf v_2,\ldots,\mathbf v_n\text{,}\) then \(\mathbf b\) is in \(\laspan{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\text{.}\)
  • If \(A\) is a \(8032\times 427\) matrix, then the span of the columns of \(A\) is a set of vectors in \(\mathbb R^{427}\text{.}\)

This exercise asks you to construct some matrices whose columns span a given set.

  • Construct a \(3\times3\) matrix whose columns span \(\mathbb R^3\text{.}\)
  • Construct a \(3\times3\) matrix whose columns span a plane in \(\mathbb R^3\text{.}\)
  • Construct a \(3\times3\) matrix whose columns span a line in \(\mathbb R^3\text{.}\)
  • Suppose that we have vectors in \(\mathbb R^8\text{,}\) \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_{10}\text{,}\) whose span is \(\mathbb R^8\text{.}\) Can every vector \(\mathbf b\) in \(\mathbb R^8\) be written as a linear combination of \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_{10}\text{?}\)
  • Suppose that we have vectors in \(\mathbb R^8\text{,}\) \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_{10}\text{,}\) whose span is \(\mathbb R^8\text{.}\) Can every vector \(\mathbf b\) in \(\mathbb R^8\) be written uniquely as a linear combination of \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_{10}\text{?}\)

span \(\mathbb R^3\text{?}\)

  • Suppose that \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n\) span \(\mathbb R^{438}\text{.}\) What can you guarantee about the value of \(n\text{?}\)
  • Can 17 vectors in \(\mathbb R^{20}\) span \(\mathbb R^{20}\text{?}\)

The following observation will be helpful in this exericse. If we want to find a solution to the equation \(AB\mathbf x = \mathbf b\text{,}\) we could first find a solution to the equation \(A\yvec = \mathbf b\) and then find a solution to the equation \(B\mathbf x = \yvec\text{.}\)

Suppose that \(A\) is a \(3\times 4\) matrix whose columns span \(\mathbb R^3\) and \(B\) is a \(4\times 5\) matrix. In this case, we can form the product \(AB\text{.}\)

  • What are the dimensions of the product \(AB\text{?}\)
  • Can you guarantee that the columns of \(AB\) span \(\mathbb R^3\text{?}\)
  • If you know additionally that the span of the columns of \(B\) is \(\mathbb R^4\text{,}\) can you guarantee that the columns of \(AB\) span \(\mathbb R^3\text{?}\)

Suppose that \(A\) is a \(12\times12\) matrix and that, for some vector \(\mathbf b\text{,}\) the equation \(A\mathbf x=\mathbf b\) has a unique solution.

  • What can you say about the pivot positions of \(A\text{?}\)
  • What can you say about the span of the columns of \(A\text{?}\)
  • If \(\mathbf c\) is some other vector in \(\mathbb R^{12}\text{,}\) what can you conclude about the equation \(A\mathbf x = \mathbf c\text{?}\)
  • What can you about the solution space to the equation \(A\mathbf x =\zerovec\text{?}\)

Suppose that

  • Is \(\mathbf v_3\) a linear combination of \(\mathbf v_1\) and \(\mathbf v_2\text{?}\) If so, find weights such that \(\mathbf v_3 = a\mathbf v_1+b\mathbf v_2\text{.}\)

can be rewritten as a linear combination of \(\mathbf v_1\) and \(\mathbf v_2\text{.}\)

  • Explain why \(\laspan{\mathbf v_1,\mathbf v_2,\mathbf v_3} = \laspan{\mathbf v_1,\mathbf v_2}\text{.}\)

As defined in this section, the span of a set of vectors is generated by taking all possible linear combinations of those vectors. This exericse will demonstrate the fact that the span can also be realized as the solution space to a linear system.

We will consider the vectors

If \(\mathbf b\) is in \(\laspan{\mathbf v_1,\mathbf v_2,\mathbf v_3}\text{,}\) then the linear system corresponding to the augmented matrix

must be consistent. This means that a pivot cannot occur in the rightmost column. Perform row operations to put this augmented matrix into a triangular form. Now identify an equation in \(a\text{,}\) \(b\text{,}\) and \(c\) that tells us when there is no pivot in the rightmost column. The solution space to this equation describes \(\laspan{\mathbf v_1,\mathbf v_2,\mathbf v_3}\text{.}\)

  • In this example, the matrix formed by the vectors \(\left[\begin{array}{rrr} \mathbf v_1& \mathbf v_2& \mathbf v_2 \\ \end{array}\right]\) has two pivot positions. Suppose we were to consider another example in which this matrix had had only one pivot position. How would this have changed the linear system describing \(\laspan{\mathbf v_1,\mathbf v_2,\mathbf v_3}\text{?}\)
  • Skip to primary navigation
  • Skip to main content
  • Skip to primary sidebar

Vector algebra: Tips and tricks to solve questions for IIT aspirants and students

Vector algebra: Tips and tricks to solve questions for IIT aspirants and students

by Cherie McCord

In the physics or technical domain, have you heard about terminologies like velocity or displacement? What are these? What does it require? These technical terminologies require magnitude and direction for the description known as the vector. In simpler language, we can describe it as the line segment with magnitude and points in a particular direction.

Therefore, like mathematics, the algebra term is also associated with vectors called vector algebra . In this, the vector quantities perform mathematical functions like addition, subtraction, division, and multiplication. Class 11th and Class 12th Science streams are unaware of these concepts. However, when it comes to solving, many students fail. It is an important concept to understand for IIT aspirants too. Therefore, we bought a few effective tips to help in solving complicated vector algebra questions.

Table of Contents

How to ace the vector algebra concept?

The correct reference.

Undoubtedly, the correct guidance will help you fetch good scores. So, why must one refer to unauthorized or low-quality learning content? For higher education, NCERT books are best. Purchase both parts of Physics and start the preparation. Remember that the NCERTs have appropriate language and proper entailed concepts

Also, referring to too many refreshers and side books for the same will confuse me. Instead, buy NCERTs or download PDFs that are available on the internet. Students with a lack of conceptual clarity will understand the module. Moreover, different competitive exams’ syllabus is based on the NCERT books only. So, we can say it is the correct reference tool.

Visualization

Vectors are related to geometry instead of algebraic concepts. Therefore, vector addition does not mean simple addition. It is the summing up of two different physical quantities. Hence, students must have the visualization power before attempting such complicated questions. Besides, concepts like linear and non-linear vectors, dot products, and others require visualization to understand better.

Parallelogram Law of Vector Addition and Triangle Law of Vector Addition are few basic concepts. These laws help in adding magnitudes where a and b denote vector quantities. So, there is a strong requirement of visualization to understand them. Moreover, these concepts require diagrams to understand. Hence, it is advisable to avail of digital learning. It contains interactive videos helping to clear concepts easily.

Memorizing formulas

Although many blogs state that rote learning is not good for academics. But, memorizing important formulas and vector results should be at your fingertips. Not in the board exam, but it will also help in competitive exams like JEE. The vector module comprises 5% in JEE exams. So, do not ignore its importance and begin the preparation from the day beginning.

Mugging up important formulas like Scalar Triple Product and Vector Triple Product will help in exams to solve questions easily. Also, it saves revision time because once the formulas are revised, students just have to implement them. Likewise, in mathematics, students must also focus on the Logarithmic functions concepts.

Practice important questions and study materials

There are several authentic publications available in digital and print format. Practice them to understand concepts. But before that, complete NCERTs. As mentioned above, it will lead to confusion and students should learn complicated concepts step by step. Authentic publications have a different range of questions that ace the knowledge level .

Moreover, previous year’s question papers are also useful resources to practice and understand concepts. Previous year question papers give a rough idea about the kind of questions that will be asked in the actual exam. It is a word of wisdom that the more you practice, the more you learn. Hence, do not stop practicing and learning new concepts.

  • Math Article

Coplanar Vectors

Coplanar vectors are the vectors which lie on the same plane, in a three-dimensional space. These are vectors which are parallel to the same plane. We can always find in a plane any two random vectors, which are coplanar. Also learn, coplanarity of two lines in a three dimensional space, represented in vector form.

Conditions for Coplanar vectors

  • If there are three vectors in a 3d-space and their scalar triple product is zero, then these three vectors are coplanar.
  • If there are three vectors in a 3d-space and they are linearly independent, then these three vectors are coplanar.
  • In case of n vectors, if no more than two vectors are linearly independent, then all vectors are coplanar.

A linear combination of vectors v 1 , …, v n with coefficients a 1 , …, a n is a vector, such that;

a 1 v 1 + … + a n v n

A linear combination a 1 v 1 + … + a n v n is called trivial if all the coefficients a 1 , …, a n is zero and if at least one of the coefficients is not zero, then it is known as non-trivial.

What are Linearly independent vectors?

The vectors, v 1 ,……v n are linearly independent if no non-trivial combination of these vectors is equal to the zero vector. That means a 1 v 1 + … + a n v n = 0 and the coefficients a 1 = 0 …, a n =0.

What are Linearly dependent vectors?

The vectors, v 1 ,……v n are linearly dependent if there exist at least one non-trivial combination of these vectors equal to zero vector.

Also, read:

  • Types of vectors
  • Vector Space

Solved Examples

Question 1: Determine whether x = {1; 2; 3}, y = {1; 1; 1}, z = {1; 2; 1} are coplanar vectors.

Solution: To check whether the three vectors x, y and z are coplanar or not, we have to calculate the scalar triple product :

x . [y × z] = (1)·(1)·(1) + (1)·(1)·(2) + (1)·(2)·(3) – (1)·(1)·(3) – (1)·(1)·(2) – (1)·(1)·(2)

= 1 + 2 + 6 – 3 – 2 – 2

As we can see, the scalar triple product is not equal to zero, hence, vectors x, y and z are not coplanar.

Question 2. If x = {1; 1; 1}, y = {1; 3; 1} and z = {2; 2; 2} are three vectors, then prove that they are coplanar.

Calculate a scalar triple product of vectors

x · [y × z] = (1)·(2)·(3) + (1)·(1)·(2) + (1)·(1)·(2) – (1)·(2)·(3) – (1)·(1)·(2) – (1)·(1)·(2)

= 6 + 2 + 2 – 6 – 2 – 2

As we can see, the scalar triple product is equal to zero, hence vectors x, y and z are coplanar.

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

how to solve questions on vector

let a=i+j+k and b=i-j+2k and c =xi+(x-2)j-k.if the vector c lies in the plane of a and b,then x equals [AIEEE 2007] a). 0 b). 1 c). _4 d). _2

i have heard that ,◼️ if a vector c is in the plane of a and b ,then c=a+constant times b ,◼️also their STP=0, But when ist method is used x equals 0,and when second method is used x equals -2 ??? please help me sorting this confusion.

Use the below formula: u,v,w are coplanar if there exist two real numbers a and b w=au+bv The answer will be x = -2

how to solve questions on vector

  • Share Share

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

close

COMMENTS

  1. Vectors

    Recognizing vectors practice Equivalent vectors Finding the components of a vector Comparing the components of vectors Practice Vectors intro 4 questions Components of vectors from endpoints 4 questions

  2. Vectors

    c = a + b c = (8, 13) + (26, 7) = (8+26, 13+7) = (34, 20) When we break up a vector like that, each part is called a component: Subtracting Vectors To subtract, first reverse the vector we want to subtract, then add. Example: subtract k = (4, 5) from v = (12, 2)

  3. Vectors Questions

    Solution: Given vector: \ (\begin {array} {l}\vec {a}= 3\hat {i}-2\hat {j}+6\hat {k}\end {array} \) Here, x = 3, y = -2 and z = 6 As we know, the formula to find the magnitude of vector A is given by: \ (\begin {array} {l}|A| = \sqrt {x^ {2}+y^ {2}+z^ {2}}\end {array} \) Now, substitute the values of x, y and z in the formula, we get

  4. Solving Problems with Vectors

    Solving Problems with Vectors We can use vectors to solve many problems involving physical quantities such as velocity, speed, weight, work and so on. Velocity: The velocity of moving object is modeled by a vector whose direction is the direction of motion and whose magnitude is the speed. Example :

  5. Solving Problems with Vectors

    Given a vector problem, a quick sketch can help you to see what's going on, and the act of transferring the problem from the written word to a diagram can often give you some insight that will help you to find a solution. Start by solving vector problems in two dimensions - it's easier to draw the diagrams - and then move on to three dimensions.

  6. Vectors and spaces

    Vectors Learn Vector intro for linear algebra Real coordinate spaces Adding vectors algebraically & graphically Multiplying a vector by a scalar Vector examples Unit vectors intro Parametric representations of lines Practice Scalar multiplication 4 questions Unit vectors 4 questions Add vectors 4 questions

  7. Vector Problems

    In order to solve vector problems: Write any information you know onto the diagram. Decide the route. Write the vector. Simplify your answer. Explain how to solve vector problems Vector problems worksheet Get your free vector problems worksheet of 20+ questions and answers. Includes reasoning and applied questions. DOWNLOAD FREE

  8. Vector word problems (practice)

    Vectors word problems Vector word problems Google Classroom You might need: Calculator Michael is running some errands. His first stop is 6 km to the east and 3 km to the south from his house. His second stop is 2 km to the west and 1 km to the south from the first stop. His third stop is 7 km to the west and 5 km to the north from the second stop.

  9. Vector Calculator

    What are the types of vectors? The common types of vectors are cartesian vectors, column vectors, row vectors, unit vectors, and position vectors. How do you add two vectors? To add two vectors, add the corresponding components from each vector. Example: the sum of (1,3) and (2,4) is (1+2,3+4), which is (3,7) Show more Related Symbolab blog posts

  10. 2.2: Vector Equations and Spans

    Definition 2.2.1 2.2. 1: Vector Equation. A vector equation is an equation involving a linear combination of vectors with possibly unknown coefficients. Note 2.2.1 2.2. 1. Asking whether or not a vector equation has a solution is the same as asking if a given vector is a linear combination of some other given vectors.

  11. 12.2: Vectors in Three Dimensions

    Learn how to extend the concept of vectors to three-dimensional space, where you can use them to describe magnitude, direction, angles, dot products, cross products, and more. This section also introduces the right-hand rule and the standard basis vectors for \(\mathbb{R}^3\). Explore examples and exercises with detailed solutions and illustrations.

  12. Vectors

    A component of a vector can be zero; Vector components can be zero. E.g. If both components of a vector are zero, this vector is known as the zero vector. Direction of the vector; Check the arrow on vector diagrams to see the direction of the vector. Check for negative signs in column vectors. Be careful calculating with negative numbers

  13. virtuallymath.com: how to solve a vector equation

    In this video you learn how to solve a basic vector equation. The key to doing this well is to treat the vectors like variables and move them around in a sim...

  14. 11.1.5 Problem Solving using Vectors

    Using trig in vector problems. When problem-solving with vectors, trigonometry can help us: convert between component form and magnitude/direction form (see Magnitude Direction); find the angle between two vectors using Cosine Rule (see Non-Right-Angled Triangles); find the area of a triangle using a variation of Area Formula (see Non-Right-Angled Triangles)

  15. Vector Equations

    We use vector equations to find out various physical quantities in 2-D or 3-D, such as velocity, acceleration, momentum, etc. Vector equations give us a diverse and more geometric way of viewing and solving the linear system of equations. Overall, we can conclude that the vector equation is: x1.t1+x2.t2+···+xk.tk = b.

  16. Vector word problem: resultant force (video)

    I could take the 3 N vector first, and take the tail of the 5 N vector at the head of the 3 N vector and shift it like this. You can add an either direction and either way you look at it, when you start at the tails and you get to the head of the second vector, you're going to have a resultant force that looks like this, which is the diagonal ...

  17. How to Solve Vector Equations

    How to Solve Vector Equations To solve vector equations, for each vector, gather the factors that are in front of it. Do this with all the vectors in the equation to make a system of equations. Theory Vector Equations A vector equation is expressed as a u → + b v → = c u → + d v →,

  18. 2.3: The span of a set of vectors

    Definition 2.3.1. The span of a set of vectors v1, v2, …, vn is the set of all linear combinations of the vectors. In other words, the span of v1, v2, …, vn consists of all the vectors b for which the equation. [v1 v2 … vn]x = b. is consistent.

  19. Vector algebra: Tips and tricks to solve questions for IIT aspirants

    The vector module comprises 5% in JEE exams. So, do not ignore its importance and begin the preparation from the day beginning. Mugging up important formulas like Scalar Triple Product and Vector Triple Product will help in exams to solve questions easily. Also, it saves revision time because once the formulas are revised, students just have to ...

  20. 1.1.3 Calculating with Vectors

    Complete the rectangle Draw the resultant vector diagonally from the origin Carefully measure the length of the resultant vector Use the scale factor to calculate the magnitude Use the protractor to measure the angle Vectors can be measured or calculated graphically if you are confident in using scales Did this video help you?

  21. Scalars and Vectors Questions

    6) Explain the fundamentals of a vector quantity. In mathematics and physics, a vector is a geometric quantity with magnitude and direction, and it is also called a spatial vector or Euclidean vector or geometric vector. A vector can be added using vector algebra. A Euclidean vector is often denoted by a directed line shape or pictorially as an ...

  22. Coplanar Vectors (Conditions & Solved Examples)

    Coplanar Vectors. Coplanar vectors are the vectors which lie on the same plane, in a three-dimensional space. These are vectors which are parallel to the same plane. We can always find in a plane any two random vectors, which are coplanar. Also learn, coplanarity of two lines in a three dimensional space, represented in vector form.

  23. Vectors and scalars questions (practice)

    Vectors and scalars questions Google Classroom Which of the following vector combinations will result in the least amount of displacement? (Note: Vectors a → , b → , d → , and e → have magnitudes double that of vectors c → and f → .) Choose 1 answer: a → − b → + e → A a → − b → + e → e → − c → + d → B e → − c → + d → a → + d → + e → C

  24. EEESAU on Instagram: "⚡ Meet Taylor Kim, General Committee Member for

    22 likes, 0 comments - eeesau.official on March 7, 2023: "⚡ Meet Taylor Kim, General Committee Member for EEESAU ⚡ First things first - what are ..."