Question

Find the equations of the bisector planes of the angles between the planes x+ 2y+ 2z= 19 and 4x− 3y+ 12z+ 3 = 0, and specify the plane that bisects the acute angle and the plane that bisects the obtuse angle.

Answer 1

The two given planes are

r.(i + 2j + 2k) = 19 (1)

and

r.(4i - 3j + 12k) + 3 = 0 (2)

The equations of the planes bisecting the angles between (1) and (2) are

Taking positive sign on the right-hand side (RHS), we get

and taking negative sign on the right-hand side, we obtain

r(25i + 7j + 62k) - 238 = 0 (4)

Hence, the two bisector planes are

r.(i + 35j - 10k) = 256

and

vector r.(25i + 17j + 62k) = 238

Now, to obtain the angle bisector bisecting the acute angle between Eqs. (1) and (2), we find the angle between one of the given planes and one of the angle bisectors. Let q be the angle between Eqs. (1) and (3), then

Thus, Eq. (3) bisects the obtuse angle between Eqs. (1) and (2) and hence Eq. (4) bisects the acute angle between the given planes.