What is the collinearity of two vectors?

Vectors and collinearity

Two non-zero vectors u and v are said to be collinear if there exists a real number k such that v = ku (or, equivalently, u = kv). In other words, one of the two vectors is a multiple of the other. Geometrically, two collinear vectors have the same direction: they point either in the same direction (if k > 0) or in opposite directions (if k < 0).

By convention, the zero vector is considered to be collinear with any vector, since 0 = 0*u for any u.

Example

Let u(2;3) and v(4;6). We note that v = 2u, since 4 = 22 and 6 = 2*3. The vectors u and v are therefore collinear.

On the other hand, if w(4;7), there is no k such that both 4 = 2k and 7 = 3k: the first equation gives k = 2, whilst the second gives k = 7/3. As 2 ≠ 7/3, u and w are not collinear.

Key points

Case	Condition	Interpretation
Collinear, same direction	v = k*u, k > 0	parallel vectors, same direction
Collinear, opposite direction	v = k*u, k < 0	parallel vectors, opposite direction
Not collinear	no suitable k	different directions

Common pitfall

Do not confuse ‘collinear’ with ‘equal’. Two collinear vectors can have completely different magnitudes (lengths); only their direction matters, not their size.