# A-level Mathematics/OCR/C4/Introduction to Vectors

< A-level Mathematics‎ | OCR‎ | C4

## Vectors as directions

Vectors symbolise directions in a space. In a coordinate system with an x and y axis, the direction from a point (x1,y1) to (x2,y2)can be represented as a vector, eg V.

In other words, to move from point from (x1,y1) to (x2,y2) would be to move in the direction of vector V

Vectors are written as column brackets, where the top row is the number of units to move in the x direction, and the bottom row is the number of units to move in the y direction. This leads to their use in describing translations, direction, movements etc, which is where many people first meet vectors.

## Position Vectors of Points

The direction from the origin (0,0) to a point (x1,y1) can be expressed as a vector, e.g., W. This is called the position vector of the point.

If a point lies on the line from (0,0) to (x1,y1) it's position vector is equal to some scalar multiple of the vector. If it were halfway from the origin to (x1,y1) it's position vector would be 1/2W

## Finding directions between points using their position vectors

This is best illustrated with an example: Point A has coordinates (xa,ya). We can say it has position vector a. Point B has coordinates (xb,yb). We can say it has position vector b.

Beginning at A, how can we find a journey using only a and b that goes between A and B?

Recall that a represents the direction from A to the origin. Similarly with b. So we could go the 'long way around' - journeying from A to B via the origin. The net result would be the same - we'd start at A and end at B.

If A is in the direction a from (0,0), the (0,0) is the opposition direction from A - i.e. -a.

So we journey backwards along a to the origin, then forwards along b to our destination B.

Our result is -a + b or b-a