Question 11
In a cartesian space with three mutually perpendicular $x$-, $y$- and $z$-axes in a counter-clockwise sense, a plane mirror has equation $x+y+3z=24$. A ray of light along the line with equation
$$\mathbf r=\begin{pmatrix}6\\1\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\0\\-4\end{pmatrix},\quad\lambda\in\mathbb R$$
hits the plane mirror and is reflected along the light with equation
$$\mathbf r=\begin{pmatrix}5\\1\\6\end{pmatrix}+v\begin{pmatrix}3\\a\\b\end{pmatrix},\quad v\in\mathbb R.$$
Find the value of $|a+b|$.
$$\mathbf r=\begin{pmatrix}6\\1\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\0\\-4\end{pmatrix},\quad\lambda\in\mathbb R$$
hits the plane mirror and is reflected along the light with equation
$$\mathbf r=\begin{pmatrix}5\\1\\6\end{pmatrix}+v\begin{pmatrix}3\\a\\b\end{pmatrix},\quad v\in\mathbb R.$$
Find the value of $|a+b|$.