Parallel vectors and collinear points

Two nonzero vectors are parallel when one is a scalar multiple of the other. A positive multiplier gives the same direction; a negative multiplier gives opposite directions. The zero vector has no direction.

To prove that distinct points \(A,B,C\) are collinear, compare vectors from a common point, such as \(\overrightarrow{AC}=k\overrightarrow{AB}\). This places \(C\) on the line through \(A\) and \(B\).

By contrast, \(\overrightarrow{AB}\parallel\overrightarrow{CD}\) does not by itself put \(A,B,C\) on one line: the two arrows may lie on separate parallel lines.

When \(\mathbf{a}\) and \(\mathbf{b}\) are nonparallel, an equality \(r\mathbf{a}+s\mathbf{b}=u\mathbf{a}+v\mathbf{b}\) gives \(r=u\) and \(s=v\). State why the two directions are nonparallel before comparing coefficients.