Three-dimensional Geometry Test

Result

Three-dimensional Geometry Test - 4

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Question 1
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Find the direction cosines of a line which makes equal angles with all three the coordinate axes.

A
±1/√2, ±1/√2, ±1/√2

B
1/√3, 1/√3, 1/√3

C
1/√2, 1/√2, 1/√2

D
±1/√3, ±1/√3, ±1/√3

Solution
Question 2
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Three planes, viz the XY Plane, XZ Plane and the YZ Plane divide the space into eight parts. Each part is called an OCTANT. What is the relation between these three planes

A
They form the angles α, β&γwith each other

B
Any two must be perpendicular to each other.

C
All three are mutually perpendicular

D
no relation between these three planes

Solution

The three mutually perpendicular coordinate planes which in turn divide the space into eight parts and each part is known as octant.
Question 3
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The co-ordinates of the vertices of the triangle are A(-2, 3, 6), B(-4, 4, 9) and C(0, 5, 8). The direction cosines of the median BE are:

A
1 , 0 , -2/3

B
3/√13, 0, -2/√13

C
3/4, 0, -2/4

D
-3/√12, 0, -2/√13

Solution
Question 4
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If the direction cosines of a line from the positive X-axis and Y-axis areThe angle of the line through Z-axis is:

A
30 °

B
45 °

C
135 °

D
60 °

Solution
Question 5
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The direction cosines of the line equally inclined with the axes are:

A
1, 1, 1

B
±1/√3, ±1/√3, ±1/√3

C
1, 0, 0

D
1/3, 1/3, 1/3

Solution
Question 6
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The direction cosines of the line whose direction ratios are 6, –6, 3 are:

A

B

C

D

Solution
Question 7
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Find the direction cosines of the x axis.

A
1, 0, 0

B
0, 0, 0

C
0, 1, 0

D
0, 0, 1

Solution

To find Direction Cosines of X-axis.
Take any two points on X-axis : A(a,0,0) &B(b,0,0)
DR of AB : (b-a,0,0)
DC of AB : ((b-a)/(((b-a)² + 0 + 0)^1/2 ), 0, 0)
: ((b-a)/(b-a) , 0 , 0)
: (1,0,0)
Question 8
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The direction cosines of the line joining the points (2, -1, 8) and (-4, -3, 5) are:

A

B

C

D

Solution

Pt. A(2, -1, 8)
Pt. B(-4, -3, 5)
Direction Ratio DR of AB : ( -4-2 , -3+1 , 5-8 )
: (-6,-2,-3)
Direction cosine of AB : ( -6/(6² +2² +3² )1/2 , -2/(6² +2² +3² )^1/2 , -3/(6² +2² +3² )^1/2 )
: ( -6/7, -2/7, -3/7)
Question 9
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What are direction numbers of a line.

A
numbers which are same as direction angles of a line

B
numbers which are proportional to the direction angles of a line

C
numbers which are proportional to the direction cosines of a line

D
none

Solution

The numbers which are proportional to direction cosines of a line are called direction numbers of the line.
Question 10
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What are direction ratios of a line.

A
numbers which are proportional to the direction angles of a line

B
numbers which are proportional to the direction cosines of a line

C
numbers which are same direction angles of a line

D
none