Intersection of Two Planes
Find the line where two non-parallel planes meet.
Method
- Find the normal vectors \(\mathbf n_1\) and \(\mathbf n_2\).
- The direction vector of the intersection line is \(\mathbf d=\mathbf n_1\times\mathbf n_2\).
- Find one point on both planes by solving the simultaneous equations.
- Write \(\mathbf r=\mathbf p+\lambda\mathbf d\).
Example structure
For planes \(x+y+z=6\) and \(2x-y+z=3\), the direction vector comes from
\[\begin{pmatrix}1\1\1\end{pmatrix}\times\begin{pmatrix}2\-1\1\end{pmatrix}.\]
Then choose a convenient variable value to find a point on the line.