Colinearity, coplanarity and linear combinations

Collinear vectors

Two vectors u and v are collinear if there exists a real number k such that v = k.u (or one of them is zero). In three-dimensional space, collinearity is tested coordinate by coordinate: the ratios x₂/x₁, y₂/y₁ and z₂/z₁ must be equal (provided the denominators are non-zero).

Example: u(2, -4, 6) and v(1, -2, 3) satisfy v = (1/2)u: they are collinear. This allows us to demonstrate that three points A, B and C lie on a straight line: A, B and C are collinear if vector(AB) and vector(AC) are collinear.

Linear combination and coplanarity

A vector w is a linear combination of u and v if there exist real numbers a and b such that w = a.u + b.v. Three non-zero vectors u, v and w in the space are coplanar if one is a linear combination of the other two, that is, if there exist real numbers a and b such that w = a·u + b·v.

Practical method

To test whether u, v and w are coplanar, we solve the system of three equations (one per coordinate) for the unknowns a and b. If the system has a solution (a, b), the vectors are coplanar; otherwise they form a basis for the space (independent).

Vectors	Coplanar?	Justification
u(1,0,0), v(0,1,0), w(2,3,0)	Yes	w = 2u + 3v
u(1,0,0), v(0,1,0), w(0,0,1)	No	inconsistent system

Common pitfall

Do not jump to the conclusion that the vectors are coplanar by checking only two out of three coordinates: the system must be consistent across ALL THREE equations simultaneously; otherwise, the calculation is incorrect.