# Column Vectors Revision and worksheets

GCSE 4 - 5AQAEdexcelOCRWJECFoundationAQA 2022Edexcel 2022OCR 2022WJEC 2022

## Column Vectors

A vector is something that has both a magnitude and direction. On diagrams they are denoted by an arrow, where the length tells us the magnitude and the arrow tells us the direction.

Vectors are often split up into two parts, which we call components: An $x$ component, which moves left or right, and a $y$ component, which moves up or down.

Make sure you are happy with the following topics before continuing:

Level 4-5 GCSE    ## What are Column Vectors?

A column vector splits the $x$ and the $y$ direction up, with $x$ on top and $y$ on bottom.

$\begin{pmatrix}x\\y \end{pmatrix}$

e.g. the vector $\mathbf{a}$ goes $3$ spaces to the right and $2$ spaces up, so would be expressed as $\begin{pmatrix}3\\2 \end{pmatrix}$. Level 4-5 GCSE    ## Adding and subtracting column vectors.

To add/subtract column vectors, we add/subtract the $x$ and $y$ values separately.

$\begin{pmatrix}a\\b \end{pmatrix} + \begin{pmatrix}c\\d \end{pmatrix} = \begin{pmatrix}a + c\\b + d \end{pmatrix}$

For example:

$\begin{pmatrix}-3\\4\end{pmatrix}+\begin{pmatrix}5\\2\end{pmatrix}=\begin{pmatrix}2\\6\end{pmatrix}$

$\begin{pmatrix}6\\3\end{pmatrix}-\begin{pmatrix}2\\1\end{pmatrix}=\begin{pmatrix}4\\2\end{pmatrix}$

Level 4-5 GCSE    ## Multiplying Column Vectors.

To multiply a column vector by a number, we multiply both values in the vector by that number, e.g.

$5\times\begin{pmatrix}2\\-3\end{pmatrix}=\begin{pmatrix}10\\-15\end{pmatrix}$

Level 4-5 GCSE    ## Example Questions

Firstly, to multiply $\mathbf{a}$ by $2$, we must multiply both of its components by $2$:

$2\mathbf{a}=2\times\begin{pmatrix}3\\8\end{pmatrix}=\begin{pmatrix}6\\16\end{pmatrix}$

Then, to add this to $\mathbf{b}$, we must add the $x$ values and $y$ values separately. Doing so, we get the answer to be:

$2\mathbf{a}+\mathbf{b}=\begin{pmatrix}6\\16\end{pmatrix}+\begin{pmatrix}-7\\2\end{pmatrix}=\begin{pmatrix}-1\\18\end{pmatrix}$

Firstly, to multiply $\mathbf{a}$ by $3$, we must multiply both of its components:

$3\mathbf{a}=3\times\begin{pmatrix}2\\7\end{pmatrix}=\begin{pmatrix}6\\21\end{pmatrix}$

Then, in order subtract $2\mathbf{b}$, we must first multiply $\mathbf{b}$ by $2$.

$2\mathbf{b}=2\times\begin{pmatrix}-5\\3\end{pmatrix}=\begin{pmatrix}-10\\6\end{pmatrix}$

Thus the calculation is:

$3\mathbf{a}-2\mathbf{b}=\begin{pmatrix}6\\21\end{pmatrix}-\begin{pmatrix}-10\\6\end{pmatrix}=\begin{pmatrix}16\\15\end{pmatrix}$

Firstly, to multiply $\mathbf{b}$ by $2$, we must multiply both of its components:

$2\mathbf{b}=2\times\begin{pmatrix}5\\-3\end{pmatrix}=\begin{pmatrix}10\\-6\end{pmatrix}$

Then, we can add $\mathbf{a}$  and  $2\mathbf{b}$:

$\mathbf{a}+2\mathbf{b}=\begin{pmatrix}6\\2\end{pmatrix}+\begin{pmatrix}10\\-6\end{pmatrix}=\begin{pmatrix}16\\-4\end{pmatrix}$

The final calculation is to subtract $\mathbf{c}$:

$\mathbf{a}+2\mathbf{b}-\mathbf{c}=\begin{pmatrix}16\\-4\end{pmatrix}-\begin{pmatrix}2\\1\end{pmatrix} = \begin{pmatrix}14\\-5\end{pmatrix}$

## Related Topics

#### Coordinates and Ratios

Level 4-5Level 6-7KS3

#### Direct and Inverse Proportion

Level 4-5Level 6-7KS3

## Worksheet and Example Questions

### (NEW) Column Vectors Exam Style Questions - MMe

Level 4-5 GCSENewOfficial MME

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