Line of intersection of planes How to find the line of intersection of the planes
$x·\begin{pmatrix}
1\\
2\\
3\\
\end{pmatrix}=0
$
and $x=\lambda_1\begin{pmatrix}
2\\
1\\
2\\
\end{pmatrix}+\lambda_2 \begin{pmatrix}
1\\
0\\
-1\\
\end{pmatrix}$. All I can figure is x is orthagonal to $\begin{pmatrix}
1\\
2\\
3\\
\end{pmatrix}$ and $x_1+2x_2+3x_3=0$, and then I'm stuck on solving those two equations to find a line of intersection. 

So each vector in the second plane looks like $\begin{bmatrix}2\lambda_1+\lambda_2\\ \lambda_1\\ 2\lambda_1-\lambda_2\end{bmatrix}$, and to be in the first plane this would have to be perpendicular to $[1,2,3]^\top$. This yields $2\lambda_1+\lambda_2+2\lambda_1+6\lambda_1-3\lambda_2=0$, hence $10\lambda_1-2\lambda_2=0$ and $\lambda_2=5\lambda_1$.
This describes the line $\lambda_1\begin{bmatrix}7\\ 1\\ -3\end{bmatrix}$.
