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

«
»

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

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=$$\frac {a1a+b1b+c1c}{\sqrt {a^2+b^2+c^2} \sqrt{a1^2+b1^2+c1^2 }}$$.

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

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

To practice Mathematics Written Test Questions for IIT JEE Exam, here is complete set of 1000+ Multiple Choice Questions and Answers.

Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs! 