Equations of Planes — Question 2

Question 2 – Solution

Strategy Cross two independent displacement vectors to obtain a normal. Compare signs after substituting test points.

Plane Here $\overrightarrow{AB}=\left\langle 2,-1,2\right\rangle$ and $\overrightarrow{AC}=\left\langle-1,2,3\right\rangle$, so $\overrightarrow{AB}\times\overrightarrow{AC}=\left\langle-7,-8,3\right\rangle$. Thus $$\boxed{7x+8y-3z=1}.$$

Side test Write $F(x,y,z)=7x+8y-3z-1$. Then $F(0,0,0)=-1$, while $F(2,1,3)=12$. The signs differ, so the points lie on $\boxed{\text{opposite sides}}$.

Verification Each of $A,B,C$ satisfies the plane equation.