Let and be two planes

MATHEMATICS · Single correct · CBSE · Mathematics · Three-Dimensional Geometry · Three-Dimensional Geometry

Problem

Let

a_1x + b_1y + c_1z + d_1 = 0

and

a_2x + b_2y + c_2z + d_2 = 0

be two planes, where

d_1, d_2 > 0

. Then, the origin lies in the acute angle if

a_1a_2 + b_1b_2 + c_1c_2 < 0

, and the origin lies in the obtuse angle if

a_1a_2 + b_1b_2 + c_1c_2 > 0

. Further, the point

(x_1, y_1, z_1)

and the origin both lie either in the acute angle or in the obtuse angle if

(a_1x_1 + b_1y_1 + c_1z_1 + d_1)(a_2x_1 + b_2y_1 + c_2z_1 + d_2) > 0

. One of

(x_1, y_1, z_1)

and the origin lies in the acute angle and the other in the obtuse angle if

(a_1x_1 + b_1y_1 + c_1z_1 + d_1)(a_2x_1 + b_2y_1 + c_2z_1 + d_2) < 0

. Given the planes

x + 2y - 3z + 5 = 0

and

2x + y + 3z + 1 = 0

. If a point

P

(2, -1, 2)

, then (1)

O

and

P

both lie in acute angle between the planes (2)

O

and

P

both lie in obtuse angle (3)

O

lies in acute angle,

P

lies in obtuse angle (4)

O

lies in obtuse angle,

P

lies in acute angle