Three-dimensional Geometry Test

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Three-dimensional Geometry Test - 10

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Question 1
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Skew lines are lines in different planes which are

A
intersecting

B
parallel and intersecting

C
neither parallel nor intersecting

D
parallel

Solution

By definition : The Skew lines are lines in different planes which are neither parallel nor intersecting .
Question 2
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If a line has the direction ratios –18, 12, –4, then what are its direction cosines ?

A

B

C

D

Solution

If a line has the direction ratios –18, 12, –4, then its direction cosines are given by:
Question 3
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Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is.

A
lx –my + nz = d

B
–lx + my + nz = d

C
lx + my + nz = d

D
lx + my + nz = –d

Solution

In Cartesian co –ordinate system Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is given by : lx + my + nz = d.
Question 4
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Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector

A

B

C

D

Solution
Question 5
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In the following case, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. 2x –y + 3z –1 = 0 and 2x –y + 3z + 3 = 0

A
The planes are perpendicular

B
The planes are at 55^∘

C
The planes are at 45^∘

D
The planes are parallel

Solution
Question 6
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Angle between skew lines is

A
the angle between two non intersecting lines drawn from any point parallel to each of the skew lines

B
the angle between two intersecting lines drawn from any point perpendicular to each of the skew lines

C
the angle between two non intersecting lines drawn from any point anti –parallel to each of the skew lines

D
the angle between two intersecting lines drawn from any point parallel to each of the skew lines

Solution

Angle between skew lines is the angle between two intersecting lines drawn from any point
Question 7
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Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector

A

B

C

D

Solution

The equation of the line which passes through the point (1, 2, 3) and is parallel to the vector Let vector and vector
Question 8
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The equation of a plane through a point whose position vector is perpendicular to the vector . is

A

B

C

D

Solution

In vector form The equation of a plane through a point whose position vector is perpendicular to the vector Is given by :
Question 9
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Find the Cartesian equation of the plane

A
x + y –z = 5

B
x + y –z = 2

C
x + y –z = 3

D
x + y –z = 4

Solution
Question 10
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In the following case, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. 4x + 8y + z –8 = 0 and y + z –4 = 0

A
45^∘

B
49^∘

C
47^∘

D
43^∘

Solution

We have, 4x + 8y + z –8 = 0 and y + z –4 = 0. Let be the angle between the planes, then