Proving collinearity with vectors
Points A, B, C are collinear if AB is a scalar multiple of AC.
Key formulas
Scalar multiple
Compute AB and AC. If one is k times the other, they share a direction and A is common, so the points line up.
Example
A(1,2), B(3,6), C(4,8). AB=(2,4), AC=(3,6)=1.5(2,4). Collinear.
Non-example
If no such k exists, the points form a triangle.
Worked example
Worked example
Problem. Are A(1,2), B(3,6) and C(4,8) collinear?
Solution. AB = (2,4), AC = (3,6) = 1.5(2,4). Yes — AC is a scalar multiple of AB, so the points are collinear.
Three midpoints
Vector methods also prove medial lines and Varignon parallelograms neatly.
Review quiz
A(0,0), B(2,2), C(5,5) are collinear because:
Practice questions
- A(0,0), B(2,2), C(5,5): collinear?
- A(0,0), B(2,1), C(3,3): collinear?
Study notes
Position vectors needed?
You can work with AB and BC only; BC = k AB is equivalent.