# Vectors

## In a nutshell

You need to know what a vector is, how to do vector arithmetic (i.e. adding, subtracting, multiplying) and also how to use vectors to construct geometric arguments and proofs.

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### Definitions: vector and scalar

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.

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## 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 $a$â€‹â€‹â€‹ steps right and $b$â€‹ steps up. |

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

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

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

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## Adding and subtracting vectors

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

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##### 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 $\textbf{a}-\textbf{b}$:*

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

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

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

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

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## Multiplying a vector with a scalar

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

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##### Example 2

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

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

â€‹$\underline{3\textbf{a}=\begin{pmatrix}12\\-3\end{pmatrix}}$â€‹â€‹

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

â€‹$\underline{-2\textbf{a}=\begin{pmatrix}-8\\2\end{pmatrix}}$â€‹â€‹

## Vector geometry: arithmetic

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

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

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

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

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

##### Example 3

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

*To get from the point $A$â€‹ to the point $B$â€‹, You go from $A$â€‹ to $O$â€‹, then from $O$â€‹ to $B$â€‹. Mathematically:*

â€‹$\overrightarrow{AB}=\overrightarrow{AO}+\overrightarrow{OB}$â€‹â€‹

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

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

*Therefore:*

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

â€‹$\underline{\overrightarrow{AB}=\textbf{b}-\textbf{a}}$â€‹â€‹

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## Vector geometry: parallel vectors and lines

### Parallel vectors

Two vectors are parallel if one is a multiple of the other.

##### Example 4

- â€‹$\begin{pmatrix}1\\2\end{pmatrix}$
*is parallel to $\begin{pmatrix}3\\6\end{pmatrix}$* because $\begin{pmatrix}3\\6\end{pmatrix}=3\times\begin{pmatrix}1\\2\end{pmatrix}$ - $\textbf{a}$â€‹
*is parallel to $-\frac{3}{2}\textbf{a}$* - â€‹$\textbf{a}+2\textbf{b}$â€‹
*is parallel to $0.5\textbf{a}+\textbf{b}$ because $\textbf{a}+2\textbf{b}=2\times(\textbf{0.5a}+\textbf{b})$*

### Straight lines

*Suppose there are three positions *$A$*â€‹, *$B$*â€‹ and *$C$*â€‹. Then, *$ABC$*â€‹ is a straight line if *$AB$*â€‹ is parallel to *$BC$*â€‹ since *$B$*â€‹ is a shared point.*

**Example 5**

*The vector $\overrightarrow{AB}$ is given to be $2\textbf{a}-3\textbf{b}$. Similarly, the vector $\overrightarrow{BC}$ = $9\textbf{b}-6\textbf{a}$. Prove that ABC is a straight line:*

*This is equivalent to showing that $\overrightarrow{AB}$ is parallel to $\overrightarrow{BC}$. To do this, show that one is a multiple of another:*

*$\overrightarrow{BC}=9\textbf{b}-6\textbf{a}=3(3\textbf{b}-2\textbf{a})=3(-2\textbf{a}+3\textbf{b})=-3(2\textbf{a}-3\textbf{b})=-3\overrightarrow{AB}$*

*$\therefore \overrightarrow{BC}=-3\times\overrightarrow{AB}$â€‹â€‹â€‹*

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*Always conclude your proof with a sentence that explains what you have done and how it answers the question:*

*Hence, the lines **$\underline{AB}$** and *$\underline{BC}$__ __*are parallel. But, since *$\underline{B}$*â€‹ is a common point, it can be deduced that *$\underline{ABC}$*â€‹ is indeed a straight line.â€‹*