Question

Find the general equation of the plane through the point (3, 2, 5) that is parallel to the plane whose general equation is 2x 3y − z = 0.

Answer 1

The plane we want and the plane we're given are parallel, so they share the same normal vector. The normal to $2x+3y-z=0$ is the vector (2, 3, -1), since $(2,3,-1)\cdot(x,y,z)=0$.

Then the plane we want has equation

$(2,3,-1) \cdot (x-3, y-2, z-5) = 0 \\\\ \implies 2(x-3) + 3(y-2) - (z-5) = 0 \\\\ \implies \boxed{2x + 3y - z = 7}$