Question

What is the difference between collinear vectors and parallel vectors?

Answer 1

Suppose that $\mathbf{v}$ and $\mathbf{w}$ are vectors in 2 -space or 3 -space with a common initial point. If one of the vectors is a scalar multiple of the other, then the vectors lie on a common line, so it is reasonable to say that they are collinear.

However, if we translate one of the vectors, then the vectors are parallel but no longer collinear.

This creates a linguistic problem because translating a vector does not change it. The only way to resolve this problem is to agree that the terms parallel and collinear mean the same thing when applied to vectors.

Although the vector $\mathbf{0}$ has no clearly defined direction, we will regard it as parallel to all vectors when convenient.