find a parametric equation for the line where two planes intersect

1. Find normal vectors for the two plane. ![[normal vector#find—vecn-of-a-plane|Find $vecn$ of a plane]]
2. The intersecting line has the same direction as $\overrightarrow{n_1}\times \overrightarrow{n_2}$. Find the cross product.
   • NOTE: You don’t have to find the unit vector (direction) of the cross product, since the magnitude doesn’t matter, only the direction and where the initial point of the vector do (which we will determine later.
3. Find a point $P$ shared by both planes where we can place the direction vector of the line.

   find a point shared by two planes

   • Case 1: the point is apparent in the plane equations.
     • $3(x+2)-2(y-1)+2(z+1)=0$
     • $(x+2)+2(y-1)-3(z+1)=0$
     • Common point: $(-2,1,-1)$
   • Case 2: try setting one variable to zero, plug it into the plane equations, then solve the resulting system of equations (two equations, two variables). See this post.

   Link to original

4. Create a parametric equation now that we have $\overrightarrow{n_1}\times \overrightarrow{n_2}$ and point $P$.

   parametric equation of line from vector and point

   Given vector $\vec{u}$ and point $(x_0,y_0,z_0)$, a parametric equation for a line can be found:

   $$\begin{array}{rl}x& =x_0+u_x\cdot t\\ y& =y_0+u_y\cdot t\\ z& =z_0+u_z\cdot t\end{array}$$Link to original