Aligning three points using vectors

Points on a Line and Collinear Vectors

Three points A, B and C lie on a line if and only if the vectors AB and AC are collinear. This follows directly from the definition: if C lies on the line (AB), then the vector AC is a multiple of the vector AB.

Method

1. Calculate the coordinates of AB and AC (always subtract the coordinates of the same starting point, in this case A).
2. Calculate the determinant xAByAC – yABxAC.
3. If the result is zero, A, B and C lie on a straight line; otherwise, they do not.

Example

Let A(1;2), B(3;6) and C(5;10). We have AB(2;4) and AC(4;8). Determinant = 28 - 44 = 16 - 16 = 0. Therefore, A, B and C lie on a straight line (we can verify that AC = 2*AB).

Another example: A(0;0), B(2;1), C(3;5). We have AB(2;1) and AC(3;5). Determinant = 25 - 13 = 10 - 3 = 7, which is not equal to 0: the points are not collinear.

Common pitfall

You must always choose the same starting point when constructing the two vectors (in this case, A). Using AB and CA, for example, without paying attention to the signs, is a frequent source of calculation errors. Stay methodical and always use the same reference point.

Key points

Vectors	Determinant	Conclusion
AB, AC collinear	= 0	A, B, C collinear
AB, AC not collinear	!= 0	A, B, C not collinear