Are these planes parallel, orthogonal, or neither(if so what is their angle of intersection?)?

Question

plane 1 : -5x - 14y +8z + 3 = 0

plane 2 : 2x + y + 3z - 2 = 0

Answer 1

The angle of intersection of the planes is the same as the angle between the normal vectors of the two planes.

-5x - 14y + 8z + 3 = 0
2x + y + 3z - 2 = 0

The normal vectors u and v are:

u = <-5, -14, 8>
v = <2, 1, 3>

Take the dot product of the vectors.

u • v = <-5, -14, 8> • <2, 1, 3> = -10 - 14 + 24 = 0

Since the dot product of the vectors is zero, the vectors, and hence the planes, are orthogonal.