Find the line of intersection between two planes.

Question

Consider the planes:

$$\begin{eqnarray}x + y + 2z = 1\\ −x + 2y + z = 5\end{eqnarray}$$

1. Find the angle between the two planes.
2. Find the line of intersection of the two planes.

I was able to answer part (1), and I found that the angle was $60$ degrees ($\pi/3$ radians). The part that I'm having trouble with is finding the line of intersection of the two planes. I have no idea where to start, any help would be much appreciated.

Answer 1

Adding both equations we get $$3y+3z=6$$ or $$y=2-z$$ substituting $z=t$ we obtain $$y=2-t$$ and $$x=2-t+2t=1$$ gives us $$x=-1-t$$ So our line has the equation $$[x,y,z]=[-1,2,0]+t[-1,-1,1]$$ where $t$ is a real number.

Answer 2

As an alternative, the normal vectors to the planes are:

• $n_1=(1,1,2)$
• $n_2=(-1,2,1)$

then a direction vector for the line is

$$n_1 \wedge n_2 = (-3,-3,3)$$