Collinearity

Objective. Prove points are collinear using scalar multiples.

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Learn: the key idea

Collinearity means three or more points lie on the same straight line, which can be proved using vectors by showing that two vectors formed from the points are scalar multiples of each other. Find two vectors sharing a common point, such as AB and AC, and if AC = kAB for some number k, the points are collinear because they point in the same or opposite direction along one line. If AB = (2, 4) and AC = (3, 6), then AC = 1.5 AB, confirming A, B and C are collinear.

Worked example

Points A, B and C have position vectors OA = (1, 2), OB = (4, 8) and OC = (7, 14). Show that A, B and C demonstrate collinearity.

1 $Find AB = OB - OA = (4-1, 8-2) = (3, 6)$
2 $Find AC = OC - OA = (7-1, 14-2) = (6, 12)$
3 $Compare: AC = (6, 12) = 2 \times (3, 6) = 2AB$
4 $Since AC is a scalar multiple of AB sharing point A, the points are collinear$

Answer: A, B and C are collinear since AC = 2AB

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Common mistake. Students sometimes conclude collinearity just because two vectors look similar in size without checking they are genuinely scalar multiples with matching components. Always verify that every component scales by the exact same factor k.

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