Mathematics Questions and Answers – Three Dimensional Geometry – Angle between a Line and a Plane

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This set of Mathematics Written Test Questions for IIT JEE Exam focuses on “Three Dimensional Geometry – Angle between a Line and a Plane”.

1. _____ is the complement of the angle between the line L and a normal line to the plane π.
a) Normal between a plane and a line
b) The angle between a line and a plane
c) Tangent between a plane and a line
d) Distance between a plane and a line

Explanation: The angle between a line and a plane is the complement of the angle between the line L and a normal line to the plane π. If θ is the angle between line whose ratios are a1, b1, c1 and the plane ax + by + cz + d = 0 then sin θ=$$\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}$$.

2. If θ is the angle between line whose ratios are a1, b1, c1 and the plane ax + by + cz + d = 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

Explanation: A mathematical symbol θ is used to find the angle between line and a normal line to the plane π along with a trigonometric function called sine. Hence, the formula
sin θ=$$\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}$$.

3. Find the angle between the planes x + 2y + 3z + 1 = 0 and (4, 1, -7).
a) – 29.34
b) 3.43
c) 11.23
d) – 17.54

Explanation: Angle between a plane and a line sin θ=$$\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}$$
sin θ = – 0.49
θ = sin-1(- 0.49)
θ = – 29.34
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4. Which trigonometric function is used to find the angle between a line and a plane?
a) Tangent
b) Cosecant
c) Secant
d) Sine

Explanation: The trigonometric function is used to find the angle between a line and a plane is sine. If θ is the angle between line whose ratios are a1, b1, c1 and the plane ax + by + cz + d = 0 then sin θ=$$\frac {a1a2.b1b2.c1c2}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}$$.

5. What is the plane equation involved in the formula sinθ=$$\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}$$?
a) a1x – b1y + c1z + d1 = 0
b) a1x2 + b1y2 + c1z2 + d1 = 0
c) ax + by + cz+ d = 0
d) a1x + b1y + c1z + d1 = 0 and ax + by + cz + d = 0

Explanation: The angle between a line and a plane is the complement of the angle between the line L and a normal line to the plane π. If θ is the angle between line whose ratios are a1, b1, c1 and the plane ax + by + cz + d = 0 then sin θ=$$\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}$$.
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6. What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are perpendicular to each other?
a) $$\frac {a1}{b1} = \frac{a2}{c1} = \frac{c2}{b2}$$
b) $$\frac {a1}{a2} = \frac{b1}{c2} = \frac{c1}{b2}$$
c) $$\frac {a}{a1} = \frac{b}{b1} = \frac{c}{c1}$$
d) $$\frac {c1}{a2} = \frac{b1}{b2} = \frac{a1}{c2}$$

Explanation: θ = 90 degrees
The relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are perpendicular to each other is $$\frac {a}{a1} = \frac{b}{b1} = \frac{c}{c1}$$.

7. What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?
a) a1a2 . b1b2 . c1c2 = 0
b) a1a + b1b + c1c = 0
c) a1a2 + b1b2 – c1c2 = 0
d) a1a2 + b1b2 – c1c2 = 0

Explanation: The relation between the plane ax + by +cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other is a1a + b1b + c1c = 0.

8. A plane and a line having an angle of 90 degrees between them are called _____
a) Orthogonal
b) Tangential
c) Normal
d) Parallel

Explanation: A plane and A line which are perpendicular to each other or a plane and a line having an angle 90 degrees between them are called orthogonal. θ is equal to 90 degrees in sin θ=$$\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}$$.

9. The condition a1a + b1b + c1c = 0 is for a plane and a line are _____ to each other.
a) integral
b) parallel
c) perpendicular
d) concentric

Explanation: The relation between the plane ax + by +cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other is a1a + b1b + c1c = 0.

10. The condition $$\frac {a}{a1} = \frac{b}{b1} = \frac{c}{c1}$$ is for a plane and a line are _____ to each other.
a) perpendicular
b) parallel
c) differential
d) tangential

Explanation: θ = 90 degrees
The relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are perpendicular to each other is $$\frac {a}{a1} = \frac{b}{b1} = \frac{c}{c1}$$.

11. Find the angle between 2x + 3y – 2z + 4 = 0 and (2, 1, 1).
a) 38.2
b) 19.64
c) 89.21
d) 29.34

Explanation: Angle between a plane and a line sin θ=$$\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}$$
sinθ = 0.49
θ = sin-1(0.49)
θ = 29.34

12. Find the angle between x + 2y + 7z + 2 = 0 and (2, 4, 6).
a) 69.69
b) 84.32
c) 66.92
d) 83.25

Explanation: Angle between a plane and a line sin θ=$$\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}$$
sinθ = 0.92
θ = 66.92

13. The plane 5x + y + kz + 1 = 0 and directional ratios of a line (3, -1, 1) are parallel, find k.
a) 4
b) -14
c) 6
d) -8

Explanation: The condition for a plane and a line are parallel to each other is a1a + b1b + c1c = 0.
5(3) + 1(-1) + k(1) = 0
K(1) = -14
K = -14

14. Find the angle between the planes 5x + 2y + 3z + 1 = 0 and (1, 1, -2).
a) 30.82
b) 3.43
c) 11.23
d) 7.54

Explanation: Angle between a plane and a line sin θ=$$\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}$$
sinθ = 0.06
θ = 3.43

15. Find k for the given plane x + 2y + kz + 2 = 0 and directional ratios of a line (8, 3, 2), if they are parallel to each other.
a) 21
b) -17
c) 12
d) -7

Explanation: The condition for a plane and a line are parallel to each other is a1a + b1b + c1c = 0.
8(1) + 3(2) + 2(k) = 0
2(k) = -14
k = -7

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