## Vectors Summary

A vector is the difference between two points. If two points are and then we write the difference (when written in a column between brackets) as a vector The arrow above means that we start from the point and end at the point If we swap and then start from and end at and write This is illustrated below. We can write points as pairs of coordinates, and vectors as columns, one number above another. If and then

$\vec{B-A}=\begin{pmatrix}7\\10\end{pmatrix}-\begin{pmatrix}3\\2\end{pmatrix}=\begin{pmatrix}4\\8\end{pmatrix}$
.
The top and bottom numbers are the distances moved in the x and y directions respectively.

The length of a vector is its length or the distance from start to end. We can find the length of a vector written in column form using Pythagoras Theorem. For A and B as above, we must go 4 in the x – direction and 8 in the y – direction. These are at right angles as shown below. Then the length of Adding and multiplying vectors is done in a natural way so that if then and  