Equation of a Plane in Different Forms

Definition of Equation of a Plane in Different Forms

Consider the locus of a point P(x, y, z). If x, y, z are allowed to vary without any restriction for their different combinations, we have a set of points like P. The surface on which these points lie, is called the locus of P. It may be a plane or any curved surface. If Q be any other point on it’s locus and all points of the straight line PQ lie on it, it is a plane. In other words if the straight line PQ, however small and in whatever direction it may be, lies completely on the locus, it is a plane, otherwise any curved surface.
 

Equation of Plane in Different Forms

Note:

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

• If a₁/a₂ = b₁/b₂ = c₁/c₂ then the planes are parallel to each other.

CXmhaU -9: Find angle between the planes 2x – y + z = 11 and x + y + 2z = 3.

hb:   

CXmhaU -10: Find the equation of the plane passing through (2, 3, –4), (1, –1, 3) and parallel to x-axis.

hb: The equation of the plane passing through (2, 3, –4) is

a(x – 2) + b(y – 3) + c(z + 4) = 0                              ……(1)

since (1, –1, 3) lie on it, we have

a + 4b – 7c = 0                                                          ……(2)

since required plane is parallel to x-axis i.e. perpendicular to YZ plane i.e.

1.a + 0.b + 0.c = 0 Þ a = 0 Þ 4b – 7c = 0 => b/7 = c/4

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

Perpendicular Distance:

The length of the perpendicular from the point P(x₁, y₁, z₁) to the plane ax + by + cz + d = 0 is|ax₁ + by₁ + cz₁ + d / √a² + b² + c²|.

Family of Planes:

Equation of plane passing through the line of intersection of two planes u = 0 and v = 0 is 
u + λv = 0.

Intersection of a Line and Plane:

If equation of a plane is ax + by + cz + d = 0, then direction cosines of normal to this plane are a, b, c. So angle between normal to the plane and a straight line having direction cosines l, m ,n is given by cos θ = al + bm + cn / √a² + b² + c².

Then angle between the plane and the straight line is π/2 – θ.

• Plane and straight line will be parallel if al + bm + cn = 0
• Plane and straight line will be perpendicular if a/l = b/m = c/n.

CXmhaU -11: Find the equation of plane passing through the intersection of planes 2x – 4y + 3z + 5= 0, x + y + z = 6 and parallel to straight line having direction cosines (1, –1, –1).

hb: Equation of required plane be

(2x – 4y + 3z + 5) + λ(x + y – z – 6) = 0

i.e. (2 + λ)x + (–4 + λ)y + z(3 – λ) + (5 – 6λ)     = 0

This plane is parallel to a straight line. So, al + bm + cn = 0

1(2 + λ) + (–1)(–4 + λ) + (–1)(3 – λ) = 0 i.e. λ = –3

∴ Equation of required plane is –x – 7y + 6z + 23 = 0

i.e. x + 7y – 6z – 23 = 0.

Bisector Planes of Angle between two Planes: 

The equation of the planes bisecting the angles between two given planes a₁x +b₁y +c₁z +d₁ = 0 and a₂x + b₂y + c₂z +d₂ = 0 is

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