Vectors

​​In a nutshell

You need to be able to recall what a vector is, and use them in two dimensions.

Definitions

Example 1

Which of these is a vector, "$$5$$​ metres" or "$$5$$ metres north"?

"$$5$$​ metres" gives magnitude, but not a direction, so it is a scalar, not a vector.

"$$5$$​ metres north" gives both magnitude and direction, so:

$$\underline{5\ metres \ north}$$ is a vector.

Representing vectors

Here are some ways to represent vectors:

The vector $$\overrightarrow{PR}$$ starts at point $$P$$ and ends at point $$R$$.

The vector can also be represented by a bold lowercase letter, for example, $$\textbf{r}$$.​

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 2

From the diagram, express the vector $$\overrightarrow{AB}$$ in terms of $$\textbf{a}$$ and $$\textbf{b}$$.​

To get from the point $$A$$ to the point $$B$$​, you go from $$A$$​ to $$O$$​, then from $$O$$​ to $$B$$:

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

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:

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

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