Equation of a line orthogonal to two other lines Consider the two lines $\ell_1:\mathbf{r}=\begin{pmatrix}1\\1\\0\end{pmatrix}+s\begin{pmatrix}-1\\2\\-2\end{pmatrix}$ and $\ell_2:\mathbf{r}=\begin{pmatrix}-2\\3\\-2\end{pmatrix}+t\begin{pmatrix}0\\-1\\2\end{pmatrix}$.  I need to determine the equation of the line $\ell_3$ whcich intersects both $\ell_1$ and $\ell_2$ at right angles.
I know that the direction of $\ell_3$ will be given by $\begin{pmatrix}2\\2\\1\end{pmatrix}$, by the cross product of the two direction vectors of the first two lines.  I am not sure how to proceed from here.

If
\begin{align}
\mathbf{r}_1&=\begin{pmatrix}1\\1\\0\end{pmatrix}+t_1\begin{pmatrix}-1\\2\\-2\end{pmatrix}\ \text{and}\\
\mathbf{r}_2&=\begin{pmatrix}-2\\3\\-2\end{pmatrix}+t_2\begin{pmatrix}0\\-1\\2\end{pmatrix}
\end{align}
are the points of intersection of $\ \ell_1\ $ and $\ \ell_2\ $ with $\ \ell_3\ $, respectively, then $\ \mathbf{r}_2-\mathbf{r}_1\ $ must be in the same direction as $\ \ell_3\ $. That is
\begin{align}
\mathbf{r}_2-\mathbf{r}_1&=\begin{pmatrix}-3\\2\\-2\end{pmatrix}-t_1\begin{pmatrix}-1\\2\\-2\end{pmatrix}+t_2\begin{pmatrix}0\\-1\\2\end{pmatrix}\\
&=t_3\begin{pmatrix}2\\2\\1\end{pmatrix}\ .
\end{align}
for some $\ t_3\ $.  Can you see how to determine the values of $\ t_1,t_2\ $and $\ t_3\ $?
