Equation of The Plane Bisecting the Angle Between Two Planes - Practice Questions & MCQ

Cartesian Form

Equation of the planes bisecting the angle between the planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 is

Proof:

Given planes are 

Let P(x, y, z) be a point on the plane bisecting the angle between planes (i) and (ii).

Let PL and PM be the length of perpendiculars from P to planes (i) and (ii).

This is equation of planes bisecting the angles between the planes (i) and (ii).

Vector Form

Equation of the planes bisecting the angle between the planes is

Bisector of the Angle between the Two Planes Containing the Origin

Let the equation of the two planes be

where d₁ and d₂ are positive.

Then the equation of the bisector of the angle between the planes (i) and (ii) containing the origin is

Bisector of the Acute and Obtuse Angle between Two Planes 

Let the equation of the two planes be

1. If a₁a₂ + b₁b₂ + c₁c₂ > 0, then the equation of the bisector of the obtuse angle is,

2. If a₁a₂ + b₁b₂ + c₁c₂ < 0, then the equation of the bisector of the obtuse angle is,