Collinear and Coplanar Vectors

Collinearity

If a vector is multiplied by a scalar k then its magnitude changes k times. The direction of such a vector remains same if k > 0 however it reverses if k < 0 and becomes indeterminate if k = 0. So by multiplying with k we obtain a vector which has same line of support as vector . These two vectors are called collinear vectors. Note here that null vector is collinear to all the vectors.

Collinearity and linear dependence

If two vectors and are collinear then they are linearly dependent and vice versa. On the other hand if two vectors are non–collinear then they are linearly independent and vice versa. In this case

Û x = 0 and y = 0

Methods to prove collinearity:

Two vectors and are collinear if there exists k ∈ R such that .
If are collinear.
Three points A(), B(), C()are collinear if there exists k∈R such that that is .
If then A, B, C are collinear.
A(), B(), C() are collinear if there exists scalars l, m, n, (not all zero) such that where l + m + n = 0.
All the above methods are equivalent and any of them can be utilized to prove the collinearity. (of 2 vectors or 3 points).

Coplanarity

Two vectors (free) are always coplanar.
Two non-collinear vectors always determine a unique plane. Hence any vector in that plane can be uniquely represented as a linear combination of these two vectors. .

(obviously vectors are coplanar).

Methods to prove coplanarity:

Three vectors are coplanar if there exists l, m∈R such that i.e., one can be uniquely expressed as a linear combination of the other two.
If are coplanar (necessary and sufficient condition).
Four points A, B, Cand Dlie in the same plane if there exist l, m∈R such that i.e. .
If = 0 then points A, B, C, D are coplanar.
A, B, C, D are coplanar if there exists scalars k, l, m, n (not all zero), such that where k + l + m + n = 0.