Vectors  can be also useful to check if three or more points A, B, C, D, ..., are collinear, that is to say, if it exists a straight line that passes by all of them. We only will consider the case of three points A, B and C, all that we will say can become widespread in more than three points.

If A, B and C are collinear, then vectors and have the same or opposite direction, that is to say, they are parallel. And if A, B and C aren't alignment, then these vectors and aren't parallel. Therefore, the condition that has to verify A, B and C in order that they are collinear is that vectors and being parallel (we remember that and are parallel if they have proportional components).

If the coordinates of A, B and C are A(a₁,a₂), B(b₁,b₂) and C(c₁,c₂), then

= (b₁ - a₁, b₂ - a₂)
= (c₁ - a₁, c₂ - a₂)

and the parallelism condition between and is: