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 abovemeans that we start from the pointand end at the point
If we swapand
then start fromand 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
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