NCERT solution Class 12 three dimensional geometry

NCERT Solutions

Class 12 Maths

Three Dimensional Geometry

NCERT Questions

Ex.11.1 Q.1
Ex.11.1 Q.2
Ex.11.1 Q.3
Ex.11.1 Q.4
Ex.11.1 Q.5
Ex.11.2 Q.1
Ex.11.2 Q.2
Ex.11.2 Q.3
Ex.11.2 Q.4
Ex.11.2 Q.5
Ex.11.2 Q.6
Ex.11.2 Q.7
Ex.11.2 Q.8
Ex.11.2 Q.9
Ex.11.2 Q.11
Ex.11.2 Q.11
Ex.11.2 Q.12
Ex.11.2 Q.13
Ex.11.2 Q.14
Ex.11.2 Q.15
Ex.11.2 Q.16
Ex.11.2 Q.17
Ex.11.3 Q.1
Ex.11.3 Q.2
Ex.11.3 Q.3
Ex.11.3 Q.4
Ex.11.3 Q.5
Ex.11.3 Q.6
Ex.11.3 Q.7
Ex.11.3 Q.8
Ex.11.3 Q.9
Ex.11.3 Q.10
Ex.11.3 Q.11
Ex.11.3 Q.12
Ex.11.3 Q.13
Ex.11.3 Q.14
Ex.Misc. Q.1
Ex.Misc.Q.2
Ex.Misc.Q.3
Ex.Misc.Q.4
Ex.Misc.Q.5
Ex.Misc.Q.6
Ex.Misc.Q.7
Ex.Misc.Q.8
Ex.Misc.Q.9
Ex.Misc.Q.10
Ex.Misc.Q.11
Ex.Misc.Q.12
Ex.Misc.Q13
Ex.Misc.Q.14
Ex.Misc.Q.15
Ex.Misc.Q.16
Ex.Misc.Q.17
Ex.Misc.Q.18
Ex.Misc.Q.19
Ex.Misc.Q.20
Ex.Misc.Q.21
Ex.Misc.Q.22
Ex.Misc.Q.23

Ex.11.3 Q.13

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

(a) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0

(b) 2x + y + 3z - 2 = 0 and x – 2y + 5 = 0

(d) 2x - y + 3z - 1 = 0 and 2x – y + 3z + 3 = 0

(e) 4x + 8y + z - 8 = 0 and y + z - 4 = 0

The direction ratios of normal to the plane L₁: a₁x + b₁y + c₁z = 0 are a₁, b₁, c₁ and L₂: a₂x + b₂y + c₂z = 0 are a₂, b₂, c₂

L₁ || L₂ if = =

L₁ Ʇ L₂ if a₁a₂ + b₁b₂ + c₁c₂= 0

The angle between L₁ and L₂ is given by,

Q = cos^-1

(a) The equations of the planes are 2x + y + 3z - 2 = 0 and x – 2y + 5 = 0

Here, a₁ = 7, b₁ = 5, c₁ = 6 and a₂ = 3, b₂ = -1, c₂ = -10

a₁a₂ + b₁b₂ + c₁c₂ = 7 * 3 + 5 * (-1) + 6 * (-10) = 21 – 5 – 60 = -44 ≠ 0

Therefore, the given planes are not perpendicular to each other.

a₁a₂ = , b₁b₂ = = -5, c₁c₂=

It can be seen that a₁a₂ ≠ b₁b₂ ≠ c₁c₂

Therefore, the given planes are not parallel to each other.

The angle between them is given by,

Q = cos^-1

=> Q = cos^-1

(b) The equations of the planes are 7x + 5y + 6z + 30 = 0 and 3x − y − 10z + 4 = 0

Here, a₁ = 2, b₁ = 1, c₁ = 3 and a₂ = 1, b₂ = -2, c₂ = 0

Now, a₁a₂ + b₁b₂ + c₁c₂ = 2 * 1 + 1 * (-2) + 3 * 0 = 2 – 2 + 0 = 0

Therefore, the given planes are perpendicular to each other.

Here, a₁ = 2, b₁ = -2, c₁ = 4 and a₂ = 3, b₂ = -3, c₂ = 6

Now, a₁a₂ + b₁b₂ + c₁c₂ = 2 * 3 + (-2) * (-3) + 4 * 6 = 6 + 6 + 24 = 36 ≠ 0

Therefore, the given planes are not perpendicular to each other.

a₁a₂ = , b₁b₂ = , c₁c₂=

It can be seen that a₁a₂ = b₁b₂ = c₁c₂

Therefore, the given planes are parallel to each other.

(d) The equations of the planes are 2x - y + 3z - 1 = 0 and 2x – y + 3z + 3 = 0

Here, a₁ = 2, b₁ = -1, c₁ = 3 and a₂ = 2, b₂ = -1, c₂ = 3

Here, a₁a₂ = = 1, b₁b₂ = , c₁c₂=

It can be seen that a₁a₂ = b₁b₂ = c₁c₂

Therefore, the given planes are parallel to each other.

(e) The equations of the planes are 4x + 8y + z - 8 = 0 and y + z - 4 = 0

Here, a₁ = 4, b₁ = 8, c₁ = 1 and a₂ = 0, b₂ = 1, c₂ = 1

Now, a₁a₂ + b₁b₂ + c₁c₂ = 4 * 0 + 8 * 1 + 1 * 1 = 0 + 8 + 1 = 9 ≠ 0

Therefore, the given planes are not perpendicular to each other.

Again, a₁a₂ = , b₁b₂ = , c₁c₂=

It can be seen that a₁a₂ ≠ b₁b₂ ≠ c₁c₂

Therefore, the given planes are not parallel to each other.

The angle between them is given by,

Q = cos^-1

=> Q = cos^-1

=> Q = 45⁰

Three Dimensional Geometry Class 12 Maths NCERT Chapter 11 NCERT Solutions

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