Mathematics Questions and Answers – Three Dimensional Geometry – Angle between Two Planes – 2

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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

Answer: b
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.
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2. If θ is the angle between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c21z + d2 = 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

Answer: b
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 a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 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

Answer: a
Explanation: The formula to find the angle between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 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.
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4. Which trigonometric function is used to find the angle between two planes?
a) Tangent
b) Cosecant
c) Secant
d) Sine
View Answer

Answer: b
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

Answer: d
Explanation: If their normals are perpendicular to each other then a1a2 + b1b2 + c1c2 = 0.
2(3) + 2(1) + s(1) = 0
s(1) = – 8
k = – 8
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6. What is the relation between the the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c21z + d2 = 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

Answer: c
Explanation: Relation between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c21z + d2 = 0, if their normal are parallel to each other is a1 : b1 : c1 = a2 : b2 : c2 ⇒ \(\frac {a1}{a2} = \frac{b1}{b2} = \frac{c1}{c2}\).

7. What is the relation between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c21z + d2 = 0, if their normal are perpendicular to each other?
a) a1a2 . b1b2 . c1c2 = 0
b) a1a2 + b1b2 + c1c2 = 0
c) a1a2 + b1b2 – c1c2 = 0
d) a1a2 + b1b2 – c1c2 = 0
View Answer

Answer: b
Explanation: θ = 90 degrees ⇒ cos θ
a1a2 + b1b2 – c1c2 = 0
Relation between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c21z + d2 = 0, if their normal are perpendicular to each other is a1a2 + b1b2 + c1c2 = 0.
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8. _____ planes have an angle 90 degrees between them.
a) Orthogonal
b) Tangential
c) Normal
d) Parallel
View Answer

Answer: a
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 a1a2 + b1b2 + c1c2 = 0 is for the planes whose normals are _____ to each other.
a) integral
b) parallel
c) perpendicular
d) concentric
View Answer

Answer: c
Explanation: θ = 90 degrees ⇒ cos θ
a1a2 + b1b2 – c1c2 = 0
Relation between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c21z + d2 = 0, if their normal are perpendicular to each other is a1a2 + b1b2 + c1c2 = 0.
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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

Answer: a
Explanation: Relation between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c21z + d2 = 0, if their normal are parallel to each other is a1 : b1 : c1 = a2 : b2 : c2 ⇒ \(\frac {a1}{a2} = \frac{b1}{b2} = \frac{c1}{c2}\).

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

Answer: d
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

Answer: c
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

Answer: b
Explanation: Relation between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c21z + d2 = 0, if their normal are perpendicular to each other is a1a2 + b1b2 + c1c2 = 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

Answer: b
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

Answer: d
Explanation: Relation between the the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c21z + d2 = 0, if their normal are perpendicular to each other is a1a2 + b1b2 + c1c2 = 0.
1(3) + 2(4) + k(-1) = 0
k(-1) = -11
k = 11

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

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