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Vectors

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 is a vector?

A vector is a quantity which has both magnitude and direction.  It can be used to show a movement.  A quantity which has just magnitude (size) is called a scalar.

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.

Vectors Image 1

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

\overrightarrow{AB} =\textbf{a}=\underline{a}

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

Vectors Image 2

\overrightarrow{BA} =-\textbf{a}=-\underline{a}

-\textbf{a}= \begin{pmatrix} \; -5 \;\\ \; -2 \; \end{pmatrix}

Step by step guide: Vector notation (coming soon)

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.

Vectors Image 3

E.g.

Vectors Image 4

-\textbf{a}= \begin{pmatrix} \; 5 \;\\ \; 2 \; \end{pmatrix}

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

-\textbf{a}= \begin{pmatrix} \; -5 \;\\ \; -2 \; \end{pmatrix}

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

Step by step guide: Column vector (coming soon)

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:

\lvert \textbf{a} \rvert = \sqrt{x^2+y^2}

E.g.

Vectors Image 5

\lvert \textbf{a} \rvert = \sqrt{x^2+y^2}=\sqrt{5^2+2^2}=\sqrt{29}

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 (coming soon)

4. Vector addition

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

Vectors Image 7

\textbf{a}= \begin{pmatrix} \; 5 \;\\ \; 2 \; \end{pmatrix}

\textbf{b}= \begin{pmatrix} \; 3 \;\\ \; -1 \; \end{pmatrix}

We can add the vectors:

Vectors Image 8

\textbf{a}+\textbf{b}= \begin{pmatrix} \; 5 \;\\ \; 2 \; \end{pmatrix} + \begin{pmatrix} \; 3 \;\\ \; -1 \; \end{pmatrix} = \begin{pmatrix} \; 8 \;\\ \; 1 \; \end{pmatrix}

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

Step by step guide: Vector addition (coming soon)

5. Vector subtraction

We can subtract vectors

When subtracting vectors, the order is important.

Vectors Image 9

\textbf{a}-\textbf{b}= \begin{pmatrix} \; 5 \;\\ \; 2 \; \end{pmatrix} - \begin{pmatrix} \; 3 \;\\ \; -1 \; \end{pmatrix} = \begin{pmatrix} \; 2 \;\\ \; 3 \; \end{pmatrix}

Step by step guide: Vector subtraction (coming soon)

6. Vector multiplication

We can multiply vectors by a scalar

E.g.

\textbf{a}= \begin{pmatrix} \; 5 \;\\ \; 2 \; \end{pmatrix}

2\textbf{a}= 2\begin{pmatrix} \; 5 \;\\ \; 2 \; \end{pmatrix} = \begin{pmatrix} \; 2 \times 5 \;\\ \; 2 \times 2 \; \end{pmatrix} = \begin{pmatrix} \; 10 \;\\ \; 4 \; \end{pmatrix}

Vectors Image 10

Step by step guide: Vector multiplication (coming soon)

7. Combining vector addition, subtraction and multiplication

Vector addition, subtraction and multiplication are often combined.

E.g.

3\textbf{a}-2\textbf{b}= 3\begin{pmatrix} \; 5 \;\\ \; 2 \; \end{pmatrix} -2 \begin{pmatrix} \; 3 \;\\ \; -1 \; \end{pmatrix} = \begin{pmatrix} \; 15 \;\\ \; 6 \; \end{pmatrix} - \begin{pmatrix} \; 6 \;\\ \; -2 \; \end{pmatrix} = \begin{pmatrix} \; 9 \;\\ \; 8 \; \end{pmatrix}

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

A scalar quantity would be speed.

Step by step guide: Vector problems (coming soon)

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

How to use vectors

Vectors worksheet

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

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Vectors worksheet

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

COMING SOON

Vectors examples

Example 1: vector notation

Here is a parallelogram. 

Vectors Example 1 Image 1

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.

Vectors Example 1 Image 2

  • Check the route – we need to start at point C and go to point O along the vectors.

Vectors Example 1 Image 4

\begin{aligned} \overrightarrow{CO}&=-\textbf{a}-\textbf{b}\\ \\ \text{or in handwriting} \\\\ \overrightarrow{CO}&=-\underline{a}-\underline{b} \end{aligned}

Example 2: column vector

Write vector \textbf{v} as a column vector

Vectors Example 2 Image 1

  • 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.

Vectors Example 2 Image 2

  • 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 .

\textbf{v}= \begin{pmatrix} \; -3 \;\\ \; 4 \; \end{pmatrix}

Example 3: magnitude of a vector

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

\textbf{c}= \begin{pmatrix} \; 5 \;\\ \; -3 \; \end{pmatrix}

Vectors Example 2 Image 3

  • We need to use Pythagoras’ theorem.

\lvert \textbf{c} \rvert = \sqrt{x^2+y^2}=\sqrt{5^2+(-3)^2}=\sqrt{34}=5.830...

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}

\textbf{a}= \begin{pmatrix} \; 4 \;\\ \; -3 \; \end{pmatrix}
\textbf{b}= \begin{pmatrix} \; 2 \;\\ \; 5 \; \end{pmatrix}

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

  • Add the x components and add the y components.

\textbf{a}+\textbf{b}= \begin{pmatrix} \; 4 \;\\ \;-3 \; \end{pmatrix} + \begin{pmatrix} \; 2 \;\\ \; 5 \; \end{pmatrix} = \begin{pmatrix} \; 4+2 \;\\ \; -3+5 \; \end{pmatrix} = \begin{pmatrix} \; 6 \;\\ \; 2 \; \end{pmatrix}

Example 5: vector subtraction

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

\textbf{v}= \begin{pmatrix} \; 3 \;\\ \; -5 \; \end{pmatrix}
\textbf{w}= \begin{pmatrix} \; -1 \;\\ \; 2 \; \end{pmatrix}

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

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

\textbf{w}-\textbf{v}= \begin{pmatrix} \; -1\;\\ \; 2 \; \end{pmatrix} - \begin{pmatrix} \; 3 \;\\ \; -5 \; \end{pmatrix} = \begin{pmatrix} \; -1-3 \;\\ \; 2--5 \; \end{pmatrix} = \begin{pmatrix} \; -4 \;\\ \; 7 \; \end{pmatrix}

Example 6: vector multiplication

Here is the vector \textbf{f}

\textbf{f}= \begin{pmatrix} \; 4 \;\\ \; -3 \; \end{pmatrix}

Work out 4 f

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

4\textbf{f}=4 \begin{pmatrix} \; 4\;\\ \; -3 \; \end{pmatrix} = \begin{pmatrix} \; 4 \times 4 \;\\ \; 4 \times -3 \; \end{pmatrix} = \begin{pmatrix} \; 16 \;\\ \; -12 \; \end{pmatrix}

Example 7: combining vector addition, subtraction and multiplication

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

\textbf{c}= \begin{pmatrix} \; 5 \;\\ \; -6 \; \end{pmatrix}
\textbf{d}= \begin{pmatrix} \; -3 \;\\ \; 1 \; \end{pmatrix}

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

  • Work out 2\textbf{c} and 4\textbf{d} separately and then subtract them.

2\textbf{c}= 2 \begin{pmatrix} \; 5 \;\\ \; -6 \; \end{pmatrix} = \begin{pmatrix} \; 10 \;\\ \; -12 \; \end{pmatrix}

4\textbf{d}= 4 \begin{pmatrix} \; -3 \;\\ \; 1 \; \end{pmatrix} = \begin{pmatrix} \; -12 \;\\ \; 4 \; \end{pmatrix}

  • So, 2\textbf{c}-4\textbf{d} would be

2\textbf{c}-4\textbf{d}= \begin{pmatrix} \; 10\;\\ \; -12 \; \end{pmatrix} - \begin{pmatrix} \; -12 \;\\ \; 4 \; \end{pmatrix} = \begin{pmatrix} \; 22 \;\\ \; -16 \; \end{pmatrix}

  • Or alternatively you can work them out in one line.

2\textbf{c}-4\textbf{d}= 2\begin{pmatrix} \; 5\;\\ \; -6 \; \end{pmatrix} - 4\begin{pmatrix} \; -3 \;\\ \; 1 \; \end{pmatrix} = \begin{pmatrix} \; 10 \;\\ \; -12 \; \end{pmatrix} - \begin{pmatrix} \; -12 \;\\ \; 4 \; \end{pmatrix} = \begin{pmatrix} \; 22 \;\\ \; -16 \; \end{pmatrix}

Common misconceptions

  • Column vectors notation

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. 

\textbf{a}+\textbf{b}=\textbf{b}+\textbf{a}

  • 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.

\textbf{a}-\textbf{b} \; \verb|≠| \; \textbf{b}-\textbf{a}

  • A component of a vector can be zero

Vector components can be zero.

E.g.

Vectors Common Misconceptions Image

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. 

E.g.

(-5)^2=25

Practice vectors questions

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

 

Vectors Practice Question 1

4\textbf{a}+2\textbf{b}
GCSE Quiz False

4\textbf{a}-2\textbf{b}
GCSE Quiz True

4\textbf{a}+12\textbf{b}
GCSE Quiz False

4\textbf{a}-12\textbf{b}
GCSE Quiz False

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

 

Vectors Practice Question 1 Explanation Image

 

\begin{aligned} \overrightarrow{BC}&=\overrightarrow{BA}+\overrightarrow{AD}+\overrightarrow{DC}\\\\ &=-7\textbf{b}+4\textbf{a}+5\textbf{b}\\\\ &=4\textbf{a}-2\textbf{b}\\\\ &\text{or in handwriting}\\\\ &=4\underline{a}-2\underline{b} \end{aligned}

2. Write this vector as a column vector:

 

Vectors Practice Question 2

\begin{pmatrix} \; -2 \;\\ \; -3 \; \end{pmatrix}
GCSE Quiz False

\begin{pmatrix} \; 2 \;\\ \; -3 \; \end{pmatrix}
GCSE Quiz False

\begin{pmatrix} \; 2 \;\\ \; 3 \; \end{pmatrix}
GCSE Quiz False

\begin{pmatrix} \; -2 \;\\ \; 3 \; \end{pmatrix}
GCSE Quiz True

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 .

 

Vectors Practice Question 2 Explanation Image

 

\begin{pmatrix} \; -2 \;\\ \; 3 \; \end{pmatrix}

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

 

\begin{pmatrix} \; -4 \;\\ \; 7 \; \end{pmatrix}

8.07
GCSE Quiz False

5.74
GCSE Quiz False

8.06
GCSE Quiz True

5.75
GCSE Quiz False

The magnitude is calculated by using Pythagoras’ theorem.

 

\begin{aligned} \sqrt{x^2+y^2}&=\sqrt{(-4)^2+7^2}\\\\ &=\sqrt{65}\\\\ &=8.0622…\\\\ &=8.06 \ \text{to 3sf} \end{aligned}

4. Work out:

 

\begin{pmatrix} \; 5 \;\\ \; -1 \; \end{pmatrix} + \begin{pmatrix} \; -6 \;\\ \; 3 \; \end{pmatrix}

\begin{pmatrix} \; 1 \;\\ \; 4 \; \end{pmatrix}
GCSE Quiz False

\begin{pmatrix} \; -1 \;\\ \; 4 \; \end{pmatrix}
GCSE Quiz False

\begin{pmatrix} \; 1 \;\\ \; 2 \; \end{pmatrix}
GCSE Quiz False

\begin{pmatrix} \; -1 \;\\ \; 2 \; \end{pmatrix}
GCSE Quiz True

Add the x components and add the y components

 

\begin{pmatrix} \; 5 \;\\ \; -1 \; \end{pmatrix} + \begin{pmatrix} \; -6 \;\\ \; 3 \; \end{pmatrix} = \begin{pmatrix} \; 5+-6 \;\\ \; -1+3 \; \end{pmatrix} = \begin{pmatrix} \; -1 \;\\ \; 2 \; \end{pmatrix}

5. Work out:

 

\begin{pmatrix} \; 4 \;\\ \; -3 \; \end{pmatrix} – \begin{pmatrix} \; 5 \;\\ \; -6 \; \end{pmatrix}

\begin{pmatrix} \; -1 \;\\ \; -3 \; \end{pmatrix}
GCSE Quiz False

\begin{pmatrix} \; -1 \;\\ \; 3 \; \end{pmatrix}
GCSE Quiz True

\begin{pmatrix} \; 1 \;\\ \; -3 \; \end{pmatrix}
GCSE Quiz False

\begin{pmatrix} \; 1 \;\\ \; 3 \; \end{pmatrix}
GCSE Quiz False
\begin{pmatrix} \; 4 \;\\ \; -3 \; \end{pmatrix} – \begin{pmatrix} \; 5 \;\\ \; -6 \; \end{pmatrix} = \begin{pmatrix} \; 4-5 \;\\ \; -3- -6 \; \end{pmatrix} = \begin{pmatrix} \; -1 \;\\ \; 3 \; \end{pmatrix}

6. Work out:

 

3\begin{pmatrix} \; 2 \;\\ \; -7 \; \end{pmatrix}

\begin{pmatrix} \; 6 \;\\ \; -7 \; \end{pmatrix}
GCSE Quiz False

\begin{pmatrix} \; 5 \;\\ \; -4 \; \end{pmatrix}
GCSE Quiz False

\begin{pmatrix} \; 6 \;\\ \; -21 \; \end{pmatrix}
GCSE Quiz True

\begin{pmatrix} \; -6 \;\\ \; 21 \; \end{pmatrix}
GCSE Quiz False

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

 

3 \begin{pmatrix} \; 2 \;\\ \; -7 \; \end{pmatrix} = \begin{pmatrix} \; 3 \times 2 \;\\ \; 3 \times -7 \; \end{pmatrix} = \begin{pmatrix} \; 6 \;\\ \; -21 \; \end{pmatrix}

Vectors GCSE questions

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

 

Vectors GCSE Question 1

 

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

 

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

 

(2 marks)

Show answer

(a)

 

\begin{pmatrix} \; 2 \;\\ \; -1 \; \end{pmatrix}

(1)

 

(b)

 

Vectors GCSE Question 1b

(1)

2. Here is trapezium

 

Vectors GCSE Question 2

 

(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.

 

(4 marks)

Show answer

(a)

 

-9\textbf{a} \ \text{(or in handwriting: } -9\underline{a})

(1)

 

(b)

 

2\textbf{b}+6\textbf{a} \ \text{(or in handwriting: } 2\underline{b}+6\underline{a})

(1)

 

(c)

 

-9\textbf{b}+2\textbf{b}+6\textbf{a} \ \text{(or in handwriting: } -9\underline{a}+2\underline{b}+6\underline{a})

(1)

 

2\textbf{b}-3\textbf{a} \ \text{(or in handwriting: } 2\underline{b}-3\underline{a})

(1)

3. Here are two vectors

 

\textbf{a}= \begin{pmatrix} \; 4 \;\\ \; -1 \; \end{pmatrix}

 

\textbf{b}= \begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix}

 

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

 

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

(3 marks)

Show answer

(a)

 

\begin{pmatrix} \; 7 \;\\ \; 1 \; \end{pmatrix}

(1)

 

(b)

 

3\textbf{a}=\begin{pmatrix} \; 12 \;\\ \; -3 \; \end{pmatrix}

 

2\textbf{b}= \begin{pmatrix} \; 6 \;\\ \; 4 \; \end{pmatrix}

(1)

 

\begin{pmatrix} \; 18 \;\\ \; 1 \; \end{pmatrix}

(1)

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

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 and this is known as a cross product.  This is also known as the vector product of two vectors.

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