## Vectors Summary

A vector is the difference between two points. If two points areandthen we write the difference(when written in a column between brackets) as a vectorThe arrow abovemeans that we start from the pointand end at the pointIf we swapandthen start fromand end at and writeThis is illustrated below.

We can write points as pairs of coordinates, and vectors as columns, one number above another. If andthen

$\vec{B-A}=\begin{pmatrix}7\\10\end{pmatrix}-\begin{pmatrix}3\\2\end{pmatrix}=\begin{pmatrix}4\\8\end{pmatrix}$
.
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 ifthen and