Equations of Planes — Question 8

Question 8

Find the distance between the parallel planes $2x-y+2z=3$ and $4x-2y+4z=21$. Find the plane exactly halfway between them.

Question 8 – Solution

Strategy Normalize the equations to identical left sides before comparing constants.

Distance Divide the second equation by 2 to get $2x-y+2z=21/2$. Therefore $$d=\frac{|21/2-3|}{\sqrt{2^2+(-1)^2+2^2}}=\frac{15/2}{3}=\boxed{5/2}.$$

Midplane Average the normalized constants: $(3+21/2)/2=27/4$. Hence $$\boxed{2x-y+2z=27/4}.$$

Verification Its constant differs from each neighboring constant by $15/4$, corresponding to distance $5/4$.