A vector is the difference between two points. If two points areandthen we can write the difference(when written in a column between brackets) asThe arrow above means that we are going from the pointto the pointIf we swapandthen we are going fromtoand writeThis is shown below.
We can also write points as pairs of coordinates and vectors as one number above another. If andthenNote 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 ifthen and