# Vectors

## In a nutshell

You need to be able to recall the definition of a vector as well as how to add, subtract, and multiply with them.

### Definitions

A **vector **is a quantity with both magnitude and direction. For example, "5 metres north" is a vector.

A **scalar** is a quantity with magnitude only. For example, "5 metres" is a scalar as you don't know in which direction.

## Representing vectors

There are different ways to represent vectors:

**Column vectors** | Vectors are commonly represented in column form. Column vectors are written like this: $\begin{pmatrix}a\\b\end{pmatrix}$. This means move $a$ steps right and $b$ steps up. |

**Bold lowercase letters** | Bold lowercase letters are used as a quick way to refer to a vector. For example, you may see $\textbf{a}$** **= $\begin{pmatrix}1\\1\end{pmatrix}$. **** |

**Uppercase letters with an arrow** | The vector from the point *A *to the point *B* can be denoted as $\overrightarrow{AB}$. |

**On a graph** | On a graph, draw vectors with an arrow to point in the direction it is moving in. |

## Adding and subtracting vectors

To add and subtract column vectors, add and subtract each row individually.

##### Example 1

*If $\textbf{a}=\begin{pmatrix}2\\1\end{pmatrix}$ and $\textbf{b}=\begin{pmatrix}-6\\3\end{pmatrix}$, work out $\textbf{a}+\textbf{b}$ and $\textbf{a}-\textbf{b}$:*

$\textbf{a}+\textbf{b}=\begin{pmatrix}2\\1\end{pmatrix}+\begin{pmatrix}-6\\3\end{pmatrix}=\begin{pmatrix}2+(-6)\\1+3\end{pmatrix}=\begin{pmatrix}-4\\4\end{pmatrix}$

$\underline{\textbf{a}+\textbf{b}=\begin{pmatrix}-4\\4\end{pmatrix}}$

$\textbf{a}-\textbf{b}=\begin{pmatrix}2\\1\end{pmatrix}-\begin{pmatrix}-6\\3\end{pmatrix}=\begin{pmatrix}2-(-6)\\1-3\end{pmatrix}=\begin{pmatrix}8\\-2\end{pmatrix}$

$\underline{\textbf{a}-\textbf{b}=\begin{pmatrix}8\\-2\end{pmatrix}}$

## Multiplying a vector with a scalar

To multiply a vector with a scalar, multiply each number in the vector by the scalar.

##### Example 2

*If $\textbf{a}=\begin{pmatrix}4\\-1\end{pmatrix}$, work out $3\textbf{a}$, and $-2\textbf{a}$:*

*$3\textbf{a}=3\begin{pmatrix}4\\-1\end{pmatrix}=\begin{pmatrix}3\times4\\3\times-1\end{pmatrix}=\begin{pmatrix}12\\-3\end{pmatrix}$*

*$\underline{3\textbf{a}=\begin{pmatrix}12\\-3\end{pmatrix}}$*

*$-2\textbf{a}=-2\begin{pmatrix}4\\-1\end{pmatrix}=\begin{pmatrix}-2\times4\\-2\times-1\end{pmatrix}=\begin{pmatrix}-8\\2\end{pmatrix}$*

$\underline{-2\textbf{a}=\begin{pmatrix}-8\\4\end{pmatrix}}$

## Vector geometry: arithmetic

Adding, subtracting, and multiplying vectors each have their own geometric interpretations.

**Addition** | Adding two vectors $\textbf{a}$ and $\textbf{b}$ gives the vector $\textbf{a}+\textbf{b}$, which represents travelling along the vector $\textbf{a}$ then along the vector $\textbf{b}$ in one journey. | |

**Multiplication by a positive scalar** | Multiplying a vector $\textbf{a}$ by a positive scalar (number) $k$ gives the vector $k\textbf{a}$, which is a vector in the same direction as $\textbf{a}$, but its length is multiplied by a factor of $k$. | |

**Negative of a vector/multiplication by **$-1$**** | The negative of a vector $\textbf{a}$ is denoted as ($-\textbf{a}$), which means it is the same length as $\textbf{a}$, but going in the opposite direction. | |

**Subtraction** | $\textbf{a}-\textbf{b}=\textbf{a}+(-\textbf{b})$. So subtracting the vector $\textbf{b}$ from $\textbf{a}$ represents going along the vector $\textbf{a}$, and then going along the vector $-\textbf{b}$. | |

##### Example 3

*In the diagram, find the vector $\overrightarrow{AB}$*:

*To get from the point $A$ to the point $B$, you go from $A$ to $O$, then from $O$ to $B$. Mathematically:*

$\overrightarrow{AB}=\overrightarrow{AO}+\overrightarrow{OB}$

*From the diagram, $\overrightarrow{OB}=\textbf{b}$. *

*If $\overrightarrow{OA}=\textbf{a}$, then $\overrightarrow{AO}=-\textbf{a}$ because it is going in the opposite direction.*

*Therefore:*

* $\overrightarrow{AB}=\overrightarrow{AO}+\overrightarrow{OB}=(-\textbf{a})+\textbf{b}=\textbf{b}-\textbf{a}$.*

*$\underline{\overrightarrow{AB}=\textbf{b}-\textbf{a}}$*