Collinearity when $\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} = \mathbf{0}$ 
Let $\mathbf{a} = \begin{pmatrix}x_a\\y_a\\z_a\end{pmatrix}$, $\mathbf{b} = \begin{pmatrix}x_b\\y_b\\z_b\end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix}x_c\\y_c\\z_c\end{pmatrix}$.
  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
  $\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} = \mathbf{0}.$

Since the cross product of two vectors gives an area, and for two vectors to give an area of $0$ they need to be on the same line (or they can be a point, but I'm assuming both are not $\mathbf 0$). However, in this problem, each of the cross products need not necessarily be $0$ since it's their sum that is $0$, and now I'm not sure what to do. 

Using that $\,\color{blue}{\mathbf{b} \times \mathbf{c} = -\,\mathbf{c} \times \mathbf{b}}\,$ and $\,\color{red}{\mathbf{a} \times \mathbf{a} = 0}\,$:
$$
\begin{align}
\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} = \mathbf{0} \;\;&\iff\;\; \mathbf{a} \times \mathbf{b} \color{blue}{- \mathbf{c} \times \mathbf{b}} + \mathbf{c} \times \mathbf{a} \color{red}{-  \mathbf{a} \times \mathbf{a}} = \mathbf{0} \\
 & \iff\;\; (\mathbf{a} - \mathbf{c}) \times \mathbf{b} + (\mathbf{c} - \mathbf{a}) \times \mathbf{a} = 0 \\
 & \iff\;\; (\mathbf{a} - \mathbf{c}) \times \mathbf{b} - (\mathbf{a} - \mathbf{c}) \times \mathbf{a} = 0 \\
 & \iff\;\; (\mathbf{a} - \mathbf{c}) \times (\mathbf{b} - \mathbf{a}) = 0
\end{align}
$$
The latter equality is equivalent to $\,\mathbf{a} - \mathbf{c}\,$ and $\, \mathbf{b} - \mathbf{a}\,$ being collinear.
