Answer
$P$, $Q$, $R$ are collinear if and only if $\overrightarrow {PQ} \times \overrightarrow {PR} = {\bf{0}}$.
Work Step by Step
Suppose three points $P$, $Q$, $R$ are collinear (lie on a line). Write ${\bf{u}} = \overrightarrow {PQ} $ and ${\bf{v}} = \overrightarrow {PR} $.
Since $P$, $Q$, $R$ are collinear, $\overrightarrow {PQ} $ is parallel to $\overrightarrow {PR} $. Thus, there exist a scalar $\lambda$ such that ${\bf{v}} = \lambda {\bf{u}}$. So, $\overrightarrow {PQ} \times \overrightarrow {PR} = {\bf{u}} \times {\bf{v}} = {\bf{u}} \times \left( {\lambda {\bf{u}}} \right) = \lambda \left( {{\bf{u}} \times {\bf{u}}} \right)$. But ${\bf{u}} \times {\bf{u}} = {\bf{0}}$. Therefore, $\overrightarrow {PQ} \times \overrightarrow {PR} = {\bf{0}}$.
Conversely, if $\overrightarrow {PQ} \times \overrightarrow {PR} = {\bf{0}}$. This implies that $\overrightarrow {PQ} $ is parallel to $\overrightarrow {PR} $. Since $P$ is the common point, $P$, $Q$, $R$ must line on the same line. That is, $P$, $Q$, $R$ are collinear.
Hence, three points $P$, $Q$, $R$ are collinear if and only if $\overrightarrow {PQ} \times \overrightarrow {PR} = {\bf{0}}$.
