Vector Geometry

This lesson covers geometric proofs using vectors — proving collinearity, parallelism, and finding intersection points — as required by the Edexcel A-Level Mathematics specification (9MA0). Vector geometry questions are among the most challenging and rewarding at A-Level.

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Proving Collinearity

Three points A, B, C are collinear if they lie on the same straight line. To prove collinearity using vectors:

Method: Show that AB = k AC (or AB = k BC) for some scalar k.

If one displacement vector is a scalar multiple of another, and they share a common point, the three points lie on the same line.

Worked Example

Show that P(2, 1, -1), Q(4, 5, 3) and R(5, 7, 5) are collinear.

PQ = (4 - 2, 5 - 1, 3 - (-1)) = (2, 4, 4) PR = (5 - 2, 7 - 1, 5 - (-1)) = (3, 6, 6)

Is PR = k × PQ? Check: 3/2 = 6/4 = 6/4 = 1.5

Yes, PR = 1.5 PQ, so P, Q, R are collinear.