This set of Mathematics Online Quiz for Engineering Entrance Exams focuses on “Three Dimensional Geometry – Angle between Two Planes – 2”.

1. _____ is the angle between the normals to two planes.

a) Normal between the planes

b) The angle between the planes

c) Tangent between the planes

d) Distance between the planes

View Answer

Explanation: The angle between the normals to two planes is called the angle between the planes. A trigonometric identity, cosine is used to find the angle called ‘θ’ between two planes.

2. If θ is the angle between the planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c2_{1}z + d_{2} = 0 then

cos θ=\(\frac {a1a2.b1b2.c1c2}{\sqrt {a1^2+b1^2+c1^2} \sqrt {a2^2+b2^2+c2^2 }}\).

a) True

b) False

View Answer

Explanation: The formula to find angle between the normal of two planes is

cos θ=\(\frac {a1a2+b1b2+c1c2}{\sqrt {a1^2+b1^2+c1^2} \sqrt{a2^2+b2^2+c2^2 }}\) not cos θ=\(\frac {a1a2.b1b2.c1c2}{\sqrt {a1^2+b1^2+c1^2} \sqrt {a2^2+b2^2+c2^2 }}\) because the numerator should contain sum of co-efficients not their product.

3. What is the formula to find the angle between the planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0?

a) cos θ=\(\frac {a1a2+b1b2+c1c2}{\sqrt {a1^2+b1^2+c1^2} \sqrt {a2^2+b2^2+c^2 }}\)

b) sec θ=\(\frac {a1a2+b1b2+c1c2}{\sqrt {a1^2+b1^2+c1^2} \sqrt{a2^2+b2^2+c2^2 }}\)

c) cos θ=\(\frac {a1a2.b1b2.c1c2}{\sqrt {a1^2+b1^2+c1^2} \sqrt{a2^2+b2^2+c2^2 }}\)

d) cot θ=\(\frac {a1a2+b1b2+c1c2}{\sqrt {a1^2+b1^2+c1^2} \sqrt{a2^2+b2^2+c2^2 }}\)

View Answer

Explanation: The formula to find the angle between the planes a

_{1}x + b

_{1}y + c

_{1}z + d

_{1}= 0 and a

_{2}x + b

_{2}y + c

_{2}z + d

_{2}= 0 is cos θ=\(\frac {a1a2+b1b2+c1c2}{\sqrt {a1^2+b1^2+c1^2} \sqrt {a2^2+b2^2+c2^2 }}\). θ is the angle between the normal of two planes.

4. Which trigonometric function is used to find the angle between two planes?

a) Tangent

b) Cosecant

c) Secant

d) Sine

View Answer

Explanation: The symbol ‘θ’ represents the angle between two planes. A trigonometric function called cosine is used the find the angle i.e.; θ between the normal of two planes.

5. Find s for the given planes 2x + 2y + sz + 2 = 0 and 3x + y + z – 2 = 0, if they are perpendicular to each other.

a) 21

b) – 7

c) 12

d) – 8

View Answer

Explanation: If their normals are perpendicular to each other then a

_{1}a

_{2}+ b

_{1}b

_{2}+ c

_{1}c

_{2}= 0.

2(3) + 2(1) + s(1) = 0

s(1) = – 8

k = – 8

6. What is the relation between the the planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c2_{1}z + d_{2} = 0, if their normal are parallel to each other?

a) \(\frac {a1}{b1} = \frac{a2}{c1} = \frac{c2}{b2}\)

b) \(\frac {a1}{a2} = \frac{b1}{c2} = \frac{c1}{b2}\)

c) \(\frac {a1}{a2} = \frac{b1}{b2} = \frac{c1}{c2}\)

d) \(\frac {c1}{a2} = \frac{b1}{b2} = \frac{a1}{c2}\)

View Answer

Explanation: Relation between the planes a

_{1}x + b

_{1}y + c

_{1}z + d

_{1}= 0 and a

_{2}x + b

_{2}y + c2

_{1}z + d

_{2}= 0, if their normal are parallel to each other is a

_{1}: b

_{1}: c

_{1}= a

_{2}: b

_{2}: c

_{2}⇒ \(\frac {a1}{a2} = \frac{b1}{b2} = \frac{c1}{c2}\).

7. What is the relation between the planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c2_{1}z + d_{2} = 0, if their normal are perpendicular to each other?

a) a_{1}a_{2} . b_{1}b_{2} . c_{1}c_{2} = 0

b) a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

c) a_{1}a_{2} + b_{1}b_{2} – c_{1}c_{2} = 0

d) a_{1}a_{2} + b_{1}b_{2} – c_{1}c_{2} = 0

View Answer

Explanation: θ = 90 degrees ⇒ cos θ

a

_{1}a

_{2}+ b

_{1}b

_{2}– c

_{1}c

_{2}= 0

Relation between the planes a

_{1}x + b

_{1}y + c

_{1}z + d

_{1}= 0 and a

_{2}x + b

_{2}y + c2

_{1}z + d

_{2}= 0, if their normal are perpendicular to each other is a

_{1}a

_{2}+ b

_{1}b

_{2}+ c

_{1}c

_{2}= 0.

8. _____ planes have an angle 90 degrees between them.

a) Orthogonal

b) Tangential

c) Normal

d) Parallel

View Answer

Explanation: The planes which are perpendicular to each other i.e.; having an angle 90 degrees between them are called orthogonal planes. Hence, Orthogonal planes have an angle 90 degrees between them.

9. The condition a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0 is for the planes whose normals are _____ to each other.

a) integral

b) parallel

c) perpendicular

d) concentric

View Answer

Explanation: θ = 90 degrees ⇒ cos θ

a

_{1}a

_{2}+ b

_{1}b

_{2}– c

_{1}c

_{2}= 0

Relation between the planes a

_{1}x + b

_{1}y + c

_{1}z + d

_{1}= 0 and a

_{2}x + b

_{2}y + c2

_{1}z + d

_{2}= 0, if their normal are perpendicular to each other is a

_{1}a

_{2}+ b

_{1}b

_{2}+ c

_{1}c

_{2}= 0.

10. The condition \(\frac {a1}{a2} = \frac{b1}{b2} = \frac{c1}{c2}\) is for the planes whose normals are _____ to each other.

a) perpendicular

b) parallel

c) differential

d) tangential

View Answer

Explanation: Relation between the planes a

_{1}x + b

_{1}y + c

_{1}z + d

_{1}= 0 and a

_{2}x + b

_{2}y + c2

_{1}z + d

_{2}= 0, if their normal are parallel to each other is a

_{1}: b

_{1}: c

_{1}= a

_{2}: b

_{2}: c

_{2}⇒ \(\frac {a1}{a2} = \frac{b1}{b2} = \frac{c1}{c2}\).

11. Find the angle between 2x + 3y – 2z + 4 = 0 and 4x + 3y + 2z + 2 = 0.

a) 38.2

b) 19.64

c) 89.21

d) 54.54

View Answer

Explanation: Angle between two planes cos cos θ=\(\frac {a1a2+b1b2+c1c2}{\sqrt {a1^2+b1^2+c1^2} \sqrt {a2^2+b2^2+c2^2 }}\)

cos θ = 0.58

θ = cos

^{-1}(0.58)

θ = 54.54

12. Find the angle between x + 2y + 7z + 2 = 0 and 4x + 4y + z + 2 = 0.

a) 69.69

b) 84.32

c) 63.25

d) 83.25

View Answer

Explanation: Angle between two planes cos θ=\(\frac {a1a2+b1b2+c1c2}{\sqrt {a1^2+b1^2+c1^2} \sqrt {a2^2+b2^2+c2^2 }}\)

cos θ = 0.45

θ = 63.25

13. The planes 5x + y + 3z + 1 = 0 and x + y – kz + 6 = 0 are orthogonal, find k.

a) 4

b) 2

c) 6

d) 8

View Answer

Explanation: Relation between the planes a

_{1}x + b

_{1}y + c

_{1}z + d

_{1}= 0 and a

_{2}x + b

_{2}y + c2

_{1}z + d

_{2}= 0, if their normal are perpendicular to each other is a

_{1}a

_{2}+ b

_{1}b

_{2}+ c

_{1}c

_{2}= 0.

5(1) + 1(1) + 3(-k) = 0

-3k = -6

K = 2

14. Find the angle between the planes 5x + y + 3z + 1 = 0 and x + y – 2z + 6 = 0.

a) 30.82

b) 34.91

c) 11.23

d) 7.54

View Answer

Explanation: Angle between two planes cos θ=\(\frac {a1a2+b1b2+c1c2}{\sqrt {a1^2+b1^2+c1^2} \sqrt {a2^2+b2^2+c2^2 }}\)

cos θ = 0.82

θ = 34.91

15. Find k for the given planes x + 2y + kz + 2 = 0 and 3x + 4y – z + 2 = 0, if they are perpendicular to each other.

a) 21

b) 17

c) 12

d) 11

View Answer

Explanation: Relation between the the planes a

_{1}x + b

_{1}y + c

_{1}z + d

_{1}= 0 and a

_{2}x + b

_{2}y + c2

_{1}z + d

_{2}= 0, if their normal are perpendicular to each other is a

_{1}a

_{2}+ b

_{1}b

_{2}+ c

_{1}c

_{2}= 0.

1(3) + 2(4) + k(-1) = 0

k(-1) = -11

k = 11

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