If $A,B,C$ are collinear, prove $\vec{A}\times \vec{B} + \vec{B}\times \vec{C} + \vec{C}\times \vec{A} =\begin{pmatrix} 0 \\ 0\\ 0 \end{pmatrix}$ Prove that if $A, B$ and $C$ are collinear, then
$\overrightarrow{A}\times \overrightarrow{B} + \overrightarrow{B}\times \overrightarrow{C} + \overrightarrow{C}\times \overrightarrow{A} =\begin{pmatrix} 0 \\ 0\\ 0 \end{pmatrix}$.
So far, I know that $\vec{AB}, \vec{BC}, \vec{AC}$ are scalar multiples and their cross product is zero. I'm not sure how to apply this to prove $\overrightarrow{A}\times \overrightarrow{B} + \overrightarrow{B}\times \overrightarrow{C} + \overrightarrow{C}\times \overrightarrow{A}$ though.

Since $A$, $B$ and $C$ are collinear, you can write $C$ as $A+\lambda(B-A)\require{cancel}$. So\begin{align}A\times B+B\times C+C\times A&=A\times B+B\times(A+\lambda(B-A))+(A+\lambda(B-A))\times A\\&=\cancel{A\times B+B\times A}+\lambda B\times(B-A)+\cancel{A\times A}+\lambda(B-A)\times A\\&=-\lambda B\times A+\lambda B\times A\\&=0.\end{align}
