A-level Mathematics/OCR/C4/Introduction to Vectors
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
To add or subtract vectors add their components line by line.
This is true for all pairs of points and is pretty useful when we start to solve problems with vectors. Of course, a diagram is often helpful to visualise things!
In Three Dimensions & beyond
Of course, most of the world around us is 3d. But all the arguments of above remain true - we just define a z axis, give points a z ordinate, and include a z component in our vectors. Similarly for higher dimensional spaces - more axis, more ordinates, more components in our vectors. But in UK A Level courses you'll only have to worry about 3 dimensions.