Intersection Between Two Planes

Example

With planes:

• $P_1:2x-2y+5z+10=0$
• $P_2:2x+y-4z+7=0$ We get:
• $\overrightarrow{n_1}=(2,-2,5)$
• $\overrightarrow{n_2}=(2,1,-4)$ The line of intersection is the solution for system of equation $⟹3y-9z-3=0$ $⟹y=3z+1$ $⟹2z+3z+1-4z+7=0$ $⟹x=\frac{z}{2}-4$ $⟹$ we get the line relations by setting $z$ as $2t$:
• $x=t-4$
• $y=6t+1$
• $z=2t$, $t\in \mathbb{R}$ $⟹$ we get the line: $(-4,1,0)+t(1,6,2)$