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:

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

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

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

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

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

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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 $$-2\textbf{a}$$:

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

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

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

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

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's going in the opposite direction.

Therefore:

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

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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

• ​$$\begin{pmatrix}1\\2\end{pmatrix}$$ is parallel to $$\begin{pmatrix}3\\6\end{pmatrix}$$ because $$\begin{pmatrix}3\\6\end{pmatrix}=3\times\begin{pmatrix}1\\2\end{pmatrix}$$
• $$\textbf{a}$$​ is parallel to $$-\frac{3}{2}\textbf{a}$$
• ​$$\textbf{a}+2\textbf{b}$$​ 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 $$A$$​, $$B$$​ and $$C$$​. Then, $$ABC$$​ is a straight line if $$AB$$​ is parallel to $$BC$$​ since $$B$$​ 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:

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

$$\therefore \overrightarrow{BC}=-3\times\overrightarrow{AB}$$​​​

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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 $$\underline{B}$$​ is a common point, it can be deduced that $$\underline{ABC}$$​ is indeed a straight line.​