ind an equation of the plane. the plane that passes through the point (1, 5, 1) and is perpendicular to the planes 4x + y − 4z = 4 and x + 8z = 5

Answer 1

1) Find the normals to the planes: <4,1,-4> and <1,0,8>

2) These can lie in the perpendicular plane, so the cross product of the normal vectors is the normal of the intersecting plane.

Find the the determinant of rows i,j,k ,<4,1,4>, and <1,0,8> = normal to plane (cross product)

3) The equation of a plane given its normal and a point on the plane if we take the normal to be <n₁,n₂,n₃> and the point is (x_p, y_p, z_p)

n₁(x - x_p) + n₂(y - y_p) + n₃(z-z_p) = 0