Parallel vectors and collinear points

Two nonzero vectors are parallel if one is a scalar multiple of the other. To prove points are collinear, compare nonzero vectors from a common point: \(\overrightarrow{AC}=k\overrightarrow{AB}\) puts \(C\) on the line through \(A,B\). Parallel arrows on separate lines do not prove collinearity.

Example

Let \(A=(1,1)\), \(B=(3,4)\), \(C=(7,10)\). Then \(\overrightarrow{AB}=\begin{pmatrix}2\\3\end{pmatrix}\) and \(\overrightarrow{AC}=\begin{pmatrix}6\\9\end{pmatrix}=3\overrightarrow{AB}\).

The vectors have the common starting point \(A\), so \(A,B,C\) are collinear. Since \(AC=3AB\) in the same direction, \(B\) is between \(A,C\) and \(AB:BC=1:2\).

When \(\mathbf a,\mathbf b\) are nonparallel, equal expressions have equal coefficients. For example, \(r\mathbf a+2\mathbf b=3\mathbf a+s\mathbf b\) gives \(r=3\), \(s=2\). State nonparallel directions before using this method; it fails when one vector is a multiple of the other.