# If $\overrightarrow a, \:\overrightarrow b,\: \overrightarrow c$ are position vectors of the points $A,B,C$ respectively, then under what condition the three points $A,B,C$ are collinear?

$\begin{array}{1 1} \overrightarrow a\times\overrightarrow b=0 \\ \overrightarrow a\times\overrightarrow b \;is\; \parallel \;to \;\overrightarrow b\times\overrightarrow c \\ \overrightarrow a\times\overrightarrow b\; is \;\perp \overrightarrow b\times\overrightarrow c \\ \overrightarrow a\times\overrightarrow b+\overrightarrow b\times\overrightarrow c+\overrightarrow c\times\overrightarrow a=0 \end{array}$

Toolbox:
• Two collinear vectors are parallel vectors.
If $A,B,C$ are collinear, then $\overrightarrow {AB}$ is || to $\overrightarrow {AC}$
$\Rightarrow\:(\overrightarrow b-\overrightarrow a)\times(\overrightarrow c-\overrightarrow a)=0$
$\Rightarrow\:\overrightarrow b\times\overrightarrow c+\overrightarrow a\times\overrightarrow b+\overrightarrow c\times\overrightarrow a=0$
edited Nov 29, 2013