Proof with 3D vectors

Let ${a} = \begin{pmatrix}x_a\\y_a\\z_a\end{pmatrix}$, ${b} = \begin{pmatrix}x_b\\y_b\\z_b\end{pmatrix}$, and ${c} = \begin{pmatrix}x_c\\y_c\\z_c\end{pmatrix}$. Show that $(x_a,y_a,z_a)$, $(x_b,y_b,z_b)$, and $(x_c,y_c,z_c)$ are collinear if and only if ${a} \times {b} + {b} \times {c} + {c} \times {a} ={0}.$

Hello,

Is there any other way to do this problem without bashing? I can't seem to find a nice and slick solution to this.

Thanks in advance!

The points are collinear iff the area of the triangle bounded by the points is zero.

$$\text{ Area } = ||(\vec a - \vec b ) \times (\vec a - \vec c ) ||$$

$$ = ||\vec a \times \vec a - \vec a \times \vec c - \vec b \times \vec a + \vec b \times \vec c ) || $$

$$ = ||0 + \vec c \times \vec a + \vec a \times \vec b + \vec b \times \vec c ) || $$