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# Vector basics

A vector is a set of instructions for moving from one point to another in three dimensions. A line which has both magnitude and direction can represent this vector.

Given two points A(1, 3, 8) and B(3, 6, 5) then the directed line segment represents a set of instructions for moving from A to B. In order to get from A to B, you need to increase the x-co-ordinates by 2, increase the y-co-ordinates by 3 and decrease the z-co-ordinates by 3.

We write .

is just one way to represent the vector

Given P(1, 4, 8) and Q(-3, 1, -4), find . To do this, think yourself into the position of point P. How many units in each direction would you have to travel to reach point Q? A quick way to do this is to subtract the values of the co-ordinates of P from the co-ordinates of Q.

Take care to subtract the right set of co-ordinates. You'd get a very different answer if you subtracted Q from P instead.

It follows that you can also work out the co-ordinates of a point if you have the co-ordinates of another point and the vector that connects them.

Given P(1, 4, 10) and is a representative of vector , find Q.

means so if P=(1, 4, 10) then Q=(3, 5, 9)

The position vector is the vector from the origin to P. If P = (3, 4, -2), say, then . is called the position vector of P. We write .

If then the length or magnitude of , written as , is given by

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