Question

Find the vector and cartesian equations of the plane passing through the point (-1, 3, 2) and perpendicular to the planes x + 2y + 2z = 5 and 3x + y + 2z = 8

Answer 1

The normal vector to the given planes x + 2y + 2z = 5 and 3x + y + 2z = 8 are respectively vector(i + 2j + 2k) and vector(3i + j + 2k). These vectors are parallel to the required plane 

The required plane passes through the point A(-1, 3, 2) and parallel to the vectors

i.e., (x + 1)(4 – 2) – (y – 3)(2 – 6) + (z – 2)(1 – 6) = 0 

2(x + 1) + 4(y – 3) – 5(z – 2) = 0 

2x + 2 + 4y – 12 – 5z + 10 = 0 

2x + 4y – 5z = 0