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Addition and subtraction

Cartesian coordinates

Negative numbers Vector notation Column vectors

This topic is relevant for:

GCSE Maths Geometry and Measure Vectors

Vector addition

Vector Addition

Here we will learn about vector addition.

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 vector addition?

Vector addition is adding one vector to another vector. This is sometimes known as a vector sum.

To do this we add the individual components of the first vector to the second vector.
First we add the horizontal components of a vector (top numbers) and then we add the vertical components of a vector (bottom numbers).

E.g.

Let’s look at some examples of vectors, vector a and vector b.

\[\textbf{a}= \left(\begin{array}{1} 3\\ 2\\ \end{array}\right) \quad \textbf{b}= \left(\begin{array}{1} 1\\ 4\\ \end{array}\right)\]

Here are the vectors represented diagrammatically:

For the addition of vectors, a + b, we add vector b onto the end of vector a.

\[\textbf{a} +\textbf{b}= \left(\begin{array}{1} 3\\ 2\\ \end{array}\right) + \left(\begin{array}{1} 1\\ 4\\ \end{array}\right)\]

\[\textbf{a} +\textbf{b}= \left(\begin{array}{1} 3\\ 2\\ \end{array}\right) + \left(\begin{array}{1} 1\\ 4\\ \end{array}\right) = \left(\begin{array}{1} 4\\ 6\\ \end{array}\right)\]

When finding a vector sum we do not need to draw a diagram. We can just add the vector components. First, add the $x$ components (top numbers) and then add the $y$ components (bottom numbers)

\[\textbf{a} +\textbf{b} = \left(\begin{array}{1} 3\\ 2\\ \end{array}\right) +\ \left(\begin{array}{1} 1\\ 4\\ \end{array}\right) = \left(\begin{array}{1} 3+1\\ 2+4\\ \end{array}\right) = \left(\begin{array}{1} 4\\ 6\\ \end{array}\right)\]

The new vector that results from adding two given vectors is known as the resultant vector.

The addition of vectors is commutative which means that the order in which we add the vectors is not important. In other words we will get the same result if we start with vector b and add vector a.

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

The diagram is a parallelogram. The resultant vector is the diagonal of the parallelogram.

What is vector addition?

How to add vectors

In order to add vectors to find the vector sum:

Add the $x$ components.
Add the $y$ components.
Write the resultant vector.

How to add vectors

Vectors worksheet (includes vector addition)

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

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Vectors worksheet (includes vector addition)

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

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Related lessons on vectors

Vector addition is part of our series of lessons to support revision on vectors. You may find it helpful to start with the main vectors lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Vector addition examples

Example 1: add vectors

Add the given vectors

\[\left(\begin{array}{1} 2\\ 4\\ \end{array}\right) \text{and}\ \left(\begin{array}{1} 5\\ 1\\ \end{array}\right)\]

Add the $x$ components.

Add the two top numbers of the original vectors

2+5=7

2Add the y components.

4+1=5

3Write the resultant vector.

\[\left(\begin{array}{1} 2\\ 4\\ \end{array}\right) + \left(\begin{array}{1} 5\\ 1\\ \end{array}\right) =\ \left(\begin{array}{1} 7\\ 5\\ \end{array}\right)\]

Example 2: add vectors

Add the given vectors

\[\left(\begin{array}{1} 4\\ 2\\ \end{array}\right) \text{and}\ \left(\begin{array}{1} 1\\ 7\\ \end{array}\right)\]

Add the $x$ components.

Add the two top numbers of the original vectors

4+1=5

Add the $y$ components.

Add the two bottom numbers

2+7=9

Write the resultant vector.

Write the two answers as a column vector

\[\left(\begin{array}{1} 4\\ 2\\ \end{array}\right) + \left(\begin{array}{1} 1\\ 7\\ \end{array}\right) =\ \left(\begin{array}{1} 5\\ 9\\ \end{array}\right)\]

Example 3: add vectors

Add the given vectors

\[\left(\begin{array}{1} 4\\ 1\\ \end{array}\right) \text{and}\ \left(\begin{array}{1} -2\\ 3\\ \end{array}\right)\]

Add the $x$ components.

Add the two top numbers of the original vectors

4+-2=2

Add the $y$ components.

Add the two bottom numbers

1+3=4

Write the resultant vector.

Write the two answers as a column vector

\[\left(\begin{array}{1} 4\\ 1\\ \end{array}\right) + \left(\begin{array}{1} -2\\ 3\\ \end{array}\right) =\ \left(\begin{array}{1} 2\\ 4\\ \end{array}\right)\]

Example 4: Add vectors

Add the given vectors

\[\left(\begin{array}{1} -5\\ 2\\ \end{array}\right) \text{and}\ \left(\begin{array}{1} 2\\ -4\\ \end{array}\right)\]

Add the $x$ components.

Add the two top numbers of the original vectors

-5+2=-3

Add the $y$ components.

Add the two bottom numbers

2+-4=-2

Write the resultant vector.

Write the two answers as a column vector

\[\left(\begin{array}{1} -5\\ 4\\ \end{array}\right) + \left(\begin{array}{1} 2\\ -4\\ \end{array}\right) =\ \left(\begin{array}{1} -3\\ -2\\ \end{array}\right)\]

Example 5: Add vectors

Add the given vectors

\[\left(\begin{array}{1} 7\\ 3\\ \end{array}\right) \text{and}\ \left(\begin{array}{1} -3\\ -4\\ \end{array}\right)\]

Add the $x$ components.

Add the two top numbers of the original vectors

7+-3=4

Add the $y$ components.

Add the two bottom numbers

3+-4=-1

Write the resultant vector.

Write the two answers as a column vector

\[\left(\begin{array}{1} 7\\ 3\\ \end{array}\right) + \left(\begin{array}{1} -3\\ -4\\ \end{array}\right) =\ \left(\begin{array}{1} 4\\ -1\\ \end{array}\right)\]

Example 6: Add vectors

Add the given vectors

\[\left(\begin{array}{1} -3\\ -1\\ \end{array}\right) \text{and}\ \left(\begin{array}{1} -2\\ 4\\ \end{array}\right)\]

Add the $x$ components.

Add the two top numbers of the original vectors

-3+-2=-5

Add the $y$ components.

Add the two bottom numbers

-1+4=3

Write the resultant vector.

Write the two answers as a column vector

\[\left(\begin{array}{1} -3\\ -1\\ \end{array}\right) + \left(\begin{array}{1} -2\\ 4\\ \end{array}\right) =\ \left(\begin{array}{1} -5\\ 3\\ \end{array}\right)\]

Common misconceptions

Adding negative integers

Mistakes can easily be made when adding a positive number and a negative number.

Make sure that negative numbers skills are well practiced.

E.g.

3+-4=3-4=-1

Column vectors notation

Column vectors only have $2$ numbers within the brackets, they have the top number and the bottom number.

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

Practice vector addition questions

\begin{pmatrix} \; 3 \;\\ \; 10 \; \end{pmatrix}

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

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

\begin{pmatrix} \; 4 \;\\ \; -3 \; \end{pmatrix}

\begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix} + \begin{pmatrix} \; 1 \;\\ \; 5 \; \end{pmatrix} = \begin{pmatrix} \; 3+1 \;\\ \; 2+5 \; \end{pmatrix} = \begin{pmatrix} \; 4 \;\\ \; 7 \; \end{pmatrix}

\begin{pmatrix} \; 5 \;\\ \; 8 \; \end{pmatrix}

\begin{pmatrix} \; 4 \;\\ \; -15 \; \end{pmatrix}

\begin{pmatrix} \; 4 \;\\ \; 2 \; \end{pmatrix}

\begin{pmatrix} \; 5 \;\\ \; 2 \; \end{pmatrix}

\begin{pmatrix} \; 4 \;\\ \; 5 \; \end{pmatrix} + \begin{pmatrix} \; 1 \;\\ \; -3 \; \end{pmatrix} = \begin{pmatrix} \; 4+1 \;\\ \; 5+-3 \; \end{pmatrix} = \begin{pmatrix} \; 5 \;\\ \; 2 \; \end{pmatrix}

\begin{pmatrix} \; 3 \;\\ \; 11 \; \end{pmatrix}

\begin{pmatrix} \; -3 \;\\ \; 11 \; \end{pmatrix}

\begin{pmatrix} \; 3 \;\\ \; 5 \; \end{pmatrix}

\begin{pmatrix} \; -3 \;\\ \; -11 \; \end{pmatrix}

\begin{pmatrix} \; -2 \;\\ \; 8 \; \end{pmatrix} + \begin{pmatrix} \; -1 \;\\ \; 3 \; \end{pmatrix} = \begin{pmatrix} \; -2+-1 \;\\ \; 8+3 \; \end{pmatrix} = \begin{pmatrix} \; -3 \;\\ \; 11 \; \end{pmatrix}

\begin{pmatrix} \; 2 \;\\ \; -4 \; \end{pmatrix}

\begin{pmatrix} \; 2 \;\\ \; 4 \; \end{pmatrix}

\begin{pmatrix} \; -2 \;\\ \; -4 \; \end{pmatrix}

\begin{pmatrix} \; -2 \;\\ \; 4 \; \end{pmatrix}

\begin{pmatrix} \; -2 \;\\ \; 3 \; \end{pmatrix} + \begin{pmatrix} \; 4 \;\\ \; -7 \; \end{pmatrix} = \begin{pmatrix} \; -2+4 \;\\ \; 3+-7 \; \end{pmatrix} = \begin{pmatrix} \; 2 \;\\ \; -4 \; \end{pmatrix}

\begin{pmatrix} \; 5 \;\\ \; 8 \; \end{pmatrix}

\begin{pmatrix} \; 6 \;\\ \; 12 \; \end{pmatrix}

\begin{pmatrix} \; 6 \;\\ \; 8 \; \end{pmatrix}

\begin{pmatrix} \; -5 \;\\ \; 8 \; \end{pmatrix}

\begin{pmatrix} \; -3 \;\\ \; 6 \; \end{pmatrix} + \begin{pmatrix} \; -2 \;\\ \; 2 \; \end{pmatrix} = \begin{pmatrix} \; -3+-2 \;\\ \; 6+2 \; \end{pmatrix} = \begin{pmatrix} \; -5 \;\\ \; 8 \; \end{pmatrix}

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

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

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

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

\begin{pmatrix} \; -4 \;\\ \; -1 \; \end{pmatrix} + \begin{pmatrix} \; -3 \;\\ \; 4 \; \end{pmatrix} = \begin{pmatrix} \; -4+-3 \;\\ \; -1+4 \; \end{pmatrix} = \begin{pmatrix} \; -7 \;\\ \; 3 \; \end{pmatrix}

Vector addition GCSE questions

1. Which of the vectors below is the sum of

$\begin{pmatrix} \; 5 \;\\ \; 3 \; \end{pmatrix} \;$ and $\; \begin{pmatrix} \; 2 \;\\ \; -5 \; \end{pmatrix}$ ?

A	$\quad \quad$	B	$\quad \quad$	C	$\quad \quad$	D
$\begin{pmatrix} \; 7 \;\\ \; 2 \; \end{pmatrix}$	$\quad \quad$	$\begin{pmatrix} \; 10 \;\\ \; -15 \; \end{pmatrix}$	$\quad \quad$	$\begin{pmatrix} \; 7 \;\\ \; -2 \; \end{pmatrix}$	$\quad \quad$	$\begin{pmatrix} \; 3 \;\\ \; 8 \; \end{pmatrix}$

(1 mark)

Show answer

(1)

2. Here are two vectors.

$\textbf{a}= \begin{pmatrix} \; -3 \;\\ \; 4 \; \end{pmatrix} \;$ and $\; \textbf{b}= \begin{pmatrix} \; 2 \;\\ \; 1 \; \end{pmatrix}$

Calculate $\textbf{a}+\textbf{b}$

(2 marks)

Show answer

\begin{pmatrix} \; -1 \;\\ \; 5 \; \end{pmatrix}

For the $x$ component (top number)

(1)

For the $y$ component (bottom number)

(1)

3. Here is a vector sum.

\begin{pmatrix} \; \text{a} \;\\ \; 3 \; \end{pmatrix} + \begin{pmatrix} \; 2 \;\\ \; \text{b} \; \end{pmatrix} = \begin{pmatrix} \; 7 \;\\ \; -1 \; \end{pmatrix}

(a) Find the value of $a$ .

(b) Find the value of $b$ .

(2 marks)

Show answer

(a)

a=5

(1)

(b)

b=-4

(1)

Learning checklist

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Add vectors

The next lessons are

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Introduction

What is vector addition?

How to add vectors

Vector addition worksheet

Vector addition examples

↓

Example 1: add vectors Example 2: add vectors Example 3: add vectors Example 4: add vectors Example 5: add vectors Example 6: add vectors

Common misconceptions

Practice vector addition questions

Vector addition GCSE questions

Learning checklist

Next lessons

Still stuck?