## Vectors 1

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

$\vec{B-A}=\begin{pmatrix}7\\10\end{pmatrix}-\begin{pmatrix}3\\2\end{pmatrix}=\begin{pmatrix}4\\8\end{pmatrix}$
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Note that the top number when a vector is written as a column vector is the distance moved in the x direction and the bottom number is the distance moved in the y direction.

The magnitude of a vector is its length or the distance from end to end. We can find the magnitude of a vector written in column form using Pythagoras theorem. For A and B as above, to get from A to B 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 We can add and multiply vectors in the natural way so that if then and  