Parallelism of straight lines and applications

Direction vectors and parallelism

A line (AB) is characterised by a direction vector AB (any vector collinear with AB is also suitable as a direction vector). Two lines (AB) and (CD) are parallel if and only if their direction vectors AB and CD are collinear.

Example

Let A(1;1), B(4;3), C(0;5), D(6;9). We have AB(3;2) and CD(6;4). Determinant = 34 - 26 = 12 - 12 = 0. Therefore, the lines (AB) and (CD) are parallel.

Special case: parallel or coincident?

If, in addition to being parallel, a point on one of the lines lies on the other (for example, if A lies on (CD)), then the two lines are in fact coincident: they are the same line, not simply two distinct parallel lines.

Application: proving that a quadrilateral is a parallelogram

To show that ABCD is a parallelogram, an effective method is to show that AB = DC, that is, that the vectors are equal (and not merely collinear). This guarantees both that the sides are parallel and that their lengths are equal.

Example: A(0;0), B(4;1), C(6;5), D(2;4). We have AB(4;1) and DC(6-2;5-4) = (4;1). Since AB = DC, ABCD is a parallelogram.

Common pitfall

Collinearity alone is not sufficient to conclude that a shape is a parallelogram: the vectors must be exactly equal (same direction and same length). Otherwise, we only have two parallel sides, not a complete parallelogram.