Angle between the Planes:

 Angle between planes can be defined as angle between normals of the planes drawn from any point to planes.

Angle between planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 is 

Note:

• If a₁a₂ +b₁b₂ +c₁c₂ = 0, then planes are perpendicular to each other.
• If   then planes are parallel to each other.

Example:  Find angle between planes 2x - y + z = 11 and x + y + 2z = 3.

Solution :     

Example: Find equation of the plane passing through (2, 3, -4), (1, -1, 3) and parallel to x-axis.

Solution :     The equation of plane passing through (2, 3, -4) is

                        a(x - 2) + b(y - 3) + c(z + 4) = 0                              ......(1)

                        as (1, -1, 3) lie on it, we have

                        a + 4b - 7c = 0                                                          ......(2)

                        as required plane is parallel to x-axis that is perpendicular to YZ plane that is

                        1.a + 0.b + 0.c = 0 => a = 0 => 4b - 7c = 0  

                        ∴ Equation of the desired plane is 7y + 4z = 5.

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