1 of 8

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

Chapter 11 -Three Dimensional Geometry

Sub Topic-Other Various Form of Equation of Plane

2 of 8

Plane passing through the intersection of two planes.
Intercept form of a plane.
Condition for co planarity of two lines.
Assignment

Outline:�

3 of 8

Plane Passing through the intersection of two planes

Let r . n₁ = d₁ ------( 1 ), r . n₂ = d₂ ----(2)

be the equations of two given planes.

Two Planes intersect in a line. Hence the points on the line are also satisfies the plane
Let a be the position vector of any point on this line. Then point A has to satisfy both the planes.
Therefore a . n₁ = d₁ and a .n₂ = d₂, for some parameter λ, the equation of the required plane is
(a . n₁ – d₁) + λ ( a . n₂ – d₂) = 0

which is in the vector form.�

BACK

4 of 8

INTERCEPT FORM OF A PLANE

Let the general form of equation of the plane be�Ax + By + Cz = D ( D ≠ 0 ) ------ ( 1 )
Consider the intercepts on x – axis, y – axis and z – axis are a, b and c respectively by the plane (1), then plane passes through the points ( a, 0, 0), (0, b, 0) and (0, 0, c)
Hence Aa + D = 0 ⇒ A = -D/a
Similarly Bb + D = 0 ⇒ B = -D/b

and Cc + D = 0 ⇒ C = -D/c

Substituting these values in the equation (1), Then the equation of the plane be

x ₊ y₊ z ₌ ₁

^{a b c}

BACK

5 of 8

Condition for Co planarity of two lines

Let the equations of two lines be

r = a₁ + λb₁---- ( 1 ) , where a₁is the position vector of a point A.

r = a₂ + μb₂ -- ( 2), where a₂ is the position vector of a point B.
Then b₁ x b₂ is perpendicular to both the lines (1) and (2).
That is b₁ x b₂ is parallel to the normal of the plane which contains lines (1) and (2)

CONTINUE

6 of 8

Continue ---

Since the require plane contains the two given lines, plane is also passing through both the points A and B.
Hence vectors AB, b₁ and b₂ are coplanar vectors.
Therefore [ AB , b₁ , b₂ ] = 0
⇒ ( a₂ – a₁) . b₁x b₂ = 0

BACK

7 of 8

Condition for Co planarity of two lines in Cartesian Form

Let x – x_{1 =}y – y₁ ₌ z – z₁ ------ ( 1 )_�a₁b₁ c₁_�and x – x₂ ₌ y – y_{2 =}z – z₂------ ( 2 )

a₂ b₂ c₂

are the equations of two lines, then condition for co planarity of two lines is

( x₂ – x₁) (y₂ – y₁) (z₂ – z₁)

a₁ b₁ c₁ = 0

a₂b₂ c₂

BACK

8 of 8

Q1.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)

ASSIGNMENTS