Question

Find an equation of the plane that passes through the point and contains the line of intersection of the given planes.
(-2, 3, 2)
x + y - z = 3 and 4x - y + 5z = 5

Answer 1

First add the two given equations of the planes, which gives 5x+4y=8. So let x=t, and this means y=2-(5/4)t and then sub in x=t and y=2-(5/4)t to get t+ 2-(5/4)t-z=3 and solving for z=-1 +(1/4)t . Note that this line of intersection contains the point P(0,2,-1) and since we know the plane also passes through Q(-2,3,2) then PQ=(-2,1,3) is a direction vector of the plane. A second direction vector of the plane is the direction vector of the line of intersection which is (1,-5/4,1/4) but let's use (4,-5,1). So, the vector equation of the required plane is (-2,3,2)+s(4,-5,1)+n(-2,1,3)