Three-dimensional Geometry Test

Result

Three-dimensional Geometry Test - 2

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Question 1
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Direction cosines of a line are

A
The tangents of the angles made by the line with the negative directions of the coordinate axes.

B
The sines of the angles made by the line with the positive directions of the coordinate axes.

C
The cotangents of the angles made by the line with the negative directions of the coordinate axes.

D
The cosines of the angles made by the line with the positive directions of the coordinate axes.

Solution

Direction cosines of a line are the cosines of the angles made by the line with the positive direction of the coordinate axis.i.e. x- axis , y-axis and z –axis respectively.
Question 2
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Shortest distance between two skew lines is

A
The line segment from origin to both the lines

B
The line segment at minimum angle to both the lines

C
The line segment parallel to both the lines

D
The line segment perpendicular to both the lines.

Solution

Shortest distance between two skew lines is The line segment perpendicular to both the lines .
Question 3
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Find the shortest distance between the lines

A
2 √29

B
2 √27

C
2 √31

D
2 √23

Solution

On comparing the given equations with :
In the cartesian form two lines

we get ;
x₁ = -1, y₁ = -1,z₁ = -1, ; a₁ = 7, b₁ = -6, c₁ = 1 and
x₂ = 3, y₂ = 5, z₂ = 7; a₂ = 1, b₂ = -2, c₂ = 1

Now the shortest distance between the lines is given by :
Question 4
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The angle θbetween the planes A₁ x + B₁ y + C₁ z + D1 = 0 and A₂ x + B₂ y + C₂ z + D₂ = 0 is given by

A

B

C

D

Solution

By definition , The angle θ between the planes A₁ x + B₁ y + C₁ z + D₁ = 0 and A₂ x + B₂ y + C₂ z + D₂ = 0 is given by :
Question 5
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Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x –y + z = 0.

A
x + z + 2 = 0

B
–x –z + 2 = 0

C
x –z + 2 = 0

D
x + z + 5 = 0

Solution

The equation of the plane through the line of intersection of the planes
Question 6
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If l, m, n are the direction cosines of a line, then

A
l² + m² + 2n² = 1.

B
l² + 2m² + n² = 1.

C
2l² + m² + n² = 1.

D
l² + m² + n² = 1.

Solution

If l, m , n are the direction cosines of a line then , we know that, l² + m² + n² = 1.
Question 7
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Shortest distance between

A

B

C

D

Solution
Question 8
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Find the shortest distance between the lines :

A

B

C

D

Solution

On comparing the given equations with:
, we get:
Question 9
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The distance of a point whose position vector is from the plane

A

B

C

D

Solution

The distance of a point whose position vector is from the plane given by :
Question 10
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Find the angle between the planes whose vector equations are

A

B

C

D

Solution