​ Vectors

Let coordinates of C be $\begin{pmatrix} 4\\ 2\\ 9 \end{pmatrix} + \lambda \begin{pmatrix}-3\\-1\\2\end{pmatrix}$ as the line passes through C.

Rewrite the coordinates of C: $\begin{pmatrix}4-3\lambda\\2-\lambda\\9+2\lambda\end{pmatrix}$.

The dot product of $\vec{\text{BA}}$ and $\vec{\text{BC}}$ is 0. (Why?)

$\begin{pmatrix}-2\\4\\6\end{pmatrix} \begin{pmatrix}-2-3\lambda&&4-\lambda&&6+2\lambda\end{pmatrix} = 0\\ -2(-2-3\lambda) + 4(4-\lambda) + 6(6+2\lambda) = 0\\ 4 + 6\lambda + 16 - 4\lambda + 36 + 12 \lambda = 0\\ 14\lambda +56 = 0\\ \lambda = -4$

So the coordinates of C is $\begin{pmatrix}16\\6\\1\end{pmatrix}$.