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Vectors

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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 \overrightarrow {AB} 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 \overrightarrow {AB} = \left( \matrix{ \;\;2 \cr \;\;3 \cr - 3 \cr} \right).

\overrightarrow {AB} is just one way to represent the vector u = \left( \matrix{ \;\;2 \cr \;\;3 \cr - 3 \cr} \right)

Given P(1, 4, 8) and Q(-3, 1, -4), find \overrightarrow {PQ}. 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.

\overrightarrow {PQ} = \left( \matrix{ - 3 - 1 \cr \;\;1 - 4 \cr - 4 - 8 \cr} \right) = \left( \matrix{ - 4 \cr - 3 \cr - 12 \cr} \right)

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 \overrightarrow {PQ} is a representative of vector u = \left( \matrix{ \;\; 2 \cr \;\; 1 \cr - 1 \cr} \right), find Q.

\left( \matrix{ \;\; 2 \cr \;\; 1 \cr - 1 \cr} \right) means \left( \matrix{ increase\;x\;by\;2 \cr increase\;y\;by\;1 \cr decrease\;z\;by - 1 \cr } \right) 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 \overrightarrow {OP} = \left( \matrix{ \;\; 3 \cr \;\; 4 \cr - 2 \cr} \right). \overrightarrow {OP} is called the position vector of P. We write p = \left( \matrix{ \;\;3 \cr \;\;4 \cr - 2 \cr} \right).

If \overrightarrow {PQ} = \left( \matrix{ \;\; 5 \cr \;\; 4 \cr - 2 \cr} \right) then the length or magnitude of \overrightarrow {PQ}, written as \left| {\overrightarrow {PQ} } \right|, is given by

\left| {\overrightarrow {PQ} } \right| = \sqrt {5^2 + 4^2 + ( - 2)^2 } = \sqrt {45}

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