Question

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
\(A \times B + B \times C + C\times A = 0.\)

I'm totally stuck on this one. By the way, the \(0\) refers to the vector \(\begin{pmatrix} 0\\0\\0 \end{pmatrix}\)

Answer 1

By collinearity,
\[
\mathbf{A}=\mathbf{d}+\lambda_1\mathbf{f}\\
\mathbf{B}=\mathbf{d}+\lambda_2\mathbf{f}\\
\mathbf{C}=\mathbf{d}+\lambda_3\mathbf{f}\\
\begin{align*}
\mathbf{A}\times \mathbf{B}&=(\mathbf{d}+\lambda_1\mathbf{f})\times(\mathbf{d}+\lambda_2\mathbf{f})\\
&=\lambda_2\mathbf{d}\times\mathbf{f}+\lambda_1\mathbf{f}\times\mathbf{d}\\
\mathbf{B}\times \mathbf{C}&=\lambda_3\mathbf{d}\times\mathbf{f}+\lambda_2\mathbf{f}\times\mathbf{d}\\
\mathbf{C}\times \mathbf{A}&=\lambda_1\mathbf{d}\times\mathbf{f}+\lambda_3\mathbf{f}\times\mathbf{d}
\end{align*}
\]
Add them together the first part follows. Let me think about the second part.