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# Collinear points

When you're working in three dimensions, the only way to prove that three points are in a line (collinear) involves showing that a common direction exists. For this, you need to use vectors.

You'll need to work through all the steps shown below if you want to gain full marks.

Here's how you would show that A(4, 1, 3), B(8, 4, 6) and C(20, 13, 15) are collinear.

First, choose two directed line segments with a common point

Express one as a multiple of the other

i.e.

and state conclusion

so and have a common direction.

Complete the proof

and have a common point. Therefore A, B and C are collinear.

Question

Show that P(0, 2, 4), Q(10, 0, 0) and R(5, 1, 2) are collinear.

i.e.

so and have a common direction

and have a common point R. Therefore P, Q and R are collinear.

When you have 3 collinear points you're often asked in what ratio one point divides the directed line segment. In the simpler case, in what ratio does the (middle) point divide the directed line segment?

Look again at the ABC example above. In what ratio does B divide AC? Make life easy for yourself by choosing two directed line segments with a common point - in this case, that's B.

then

i.e. AB:BC = 1:3

BBC iD