�Mathematics –Class XII��Unit IV-Vector & 3D

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Chapter 11 -Three Dimensional Geometry

Sub Topic- Angle Between Two Planes

• Angle between two planes.
• Distance of a point from a plane.
• Angle between a plane and a line.
• Assignment

Outline:�

Angle Between Two Planes

• Let r . n₁ = d₁ and

r . n₂ = d₂ be the equations of two planes which intersect� at an angle θ as shown in the diagram.

π₂

C

CONTINUE

Continue

• Angle between two planes= angle between their normals. Now,
• n₁ . n₂ = |n₁| |n₂| Cos θ

Hence

( n₁ . n₂)

Cos θ = ----------

| n₁| | n₂ |

BACK

Distance of a point from a plane

• Let P be a given point with the position vector a and a given plane�π₁: r .n = d –(1)
• Draw PN perpendicular to the given plane.

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P( a )

N

CONTINUE

Cartesian Form of Distance of a Point from a given Plane

• Let P(x₁, y₁, z₁) be the given point and Ax + By + Cz = D be the given plane.
• Then the distance of the point ‘P’ from the given plane is MN = ‘d’

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Ax₁+By₁+ Cz₁ - D

• d = ------------------------

| √ A² + B² + C² |

BACK

• Draw a plane π₂ passing through point ‘P’ and parallel to plane π₁ .
• Then the normal of the plane π₁ is also parallel to plane π₂
• Hence the equation of the plane is π₂ � ( r – a ) . n = 0�⇒ r . n = a . n ----(2)
• MN = OM – ON� = a . n - d

O

N

M

P

π₁

π₂

CONTINUE

Angle between a Line an a Plane

• Let r = a + λ b represents a line and r . n = d , represents a plane.
• Let θ be the angle between a line and a plane and ϕ be the angle between line and normal.
• From triangle ABC, AB is the normal of the plane.
• Cos ϕ = ( b . n ) / (| b | .| n | )
• ϕ = 90⁰ - θ ⇒ Cos ϕ = Sin θ
• Sin θ = ( b . n ) / (| b | .| n | )

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A

B

C

θ

ϕ

BACK

Q1.Show that the following planes are perpendicular to each other.�x – y + z – 2 = 0 and 3x + 2y – z + 4 = 0

Q2.Find the equation of the plane containing the line,

x – 4 ₌ y – 3 ₌ z – 2

1 4 5

and passing through the line

x – 3 ₌ y – 2 ₌ z

1 -4 5

Q3.Find the equation of the plane passing through the intersection of planes �x + y + z – 6 = 0 , 2x + 3y + 4z + 5 = 0 and passing through the point ( 1, 1, 1)