Three Dimensional Geometry Test

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Three Dimensional Geometry Test - 4

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Question 1
2 / -0.83

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
Question 11
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Find the direction cosines of the side AB of the triangle whose vertices are A(3, 5, -4), B(-1, 1, 2) and C(-5, -5, -2)

A

B

C

D

Solution
Question 12
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If l, m , n are the direction cosines of any line, then sum of the squares of the direction cosines of the line is always

A
-1

B
√3

C
1

D
0

Solution

Let OP be any line through the origin O which has direction cosines l, m, n.
Let P be the point having coordinates (x, y, z) and OP = r.
Then OP ²= x ²+ y ²+ z ²= r ²... (1)
From P draw PA, PB, PC perpendicular on the coordinate axes, so that
OA = x, OB = y, OC = z.
Also, ∠POA = α, ∠POB = β, ∠POC = γ.
From triangle AOP, l = cos α= x / r ⇒x = lr.
Similarly, y = mr, z = nr.
Adding all we get x ²+ y ²+ z ²= r ²(l ²+ m ²+ n ²).
⇒r ²= r ²(l ²+ m ²+ n ²)
⇒l ²+ m ²+ n ²= 1.
Question 13
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If a line has the direction ratios -4, 18, -12 then what are its direction cosines?

A
-2, 9, -6

B
-4, 18, -12

C
2/11, 9/11, 6/11

D
-2/11, 9/11, -6/11

Solution

DR of the line : (-4, 18 -12)
DC of the line : (-4/k, 18/k, -12/k)
where k = ((4² ) + (18² ) + (12)² )^1/2
= (16 + 324 + 144)^1/2
= (484)^1/2
= 22
So, DC : (-4/22, 18/22, -12/22)
: (-2/11 , 9/11 , -6/11)
Question 14
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The direction cosines of the line equally inclined with the axes, are:

A
1, 1, 1,

B
0, 0, 0

C
1/3, 1/3, 1/3

D
1/√3, 1/√3, 1/√3

Solution

Cos^2 alpha+cos ^2 beta +cos^ 2 gamma =1
put alpha =beta. =gamma
we get cosalpha= 1/√3
DC's of given line (1/√3,1/√3,1/3)
Question 15
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If a line makes angles 45 °,150 °, 135 °, with x, y and z-axes respectively, find its direction cosines.

A

B

C

D

Solution
Question 16
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Find the equation of the set of points which are equidistant from the points (1, 2 , 3) and (3, 2, -1)

A
x + 2z = 0

B
y + 2z = 0

C
x –2z = 0

D
x –2y = 0

Solution

Pt. A(1, 2 , 3)
Pt. B(3, 2, -1)
Let P(x,y,z)
So, AP = BP
((x-1)² + (y-2)² + (z-3)² )^1/2 = ((x-3)² + (y-2)² + (z+1)² )^1/2
(x-1)² + (y-2)² + (z-3)² ) = (x-3)² + (y-2)² + (z+1)²
x² +1 -2x + y² + 4 - 4y + z² + 9 –6z = x² +9 -6x + y² + 4 - 4y + z² + 1 + 2z
4x –8z = 0
x –2z = 0
Question 17
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If a line in the ZX-plane makes an angle 60o with Z-axis, the direction cosines of this line are:

A

B

C

D

Solution
Question 18
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A line makes angles α, β, γwith the positive directions of X-axis, Y-axis and Z-axis, respectively, then the directions cosines of the line are:

A
180 °- α, 180 °- β, 180 °- γ

B
cos - α, cos - β, cos - γ

C
90 °- α, 90 °- β, 90 °- γ

D
sin α, sin β, sin γ

Solution

cos α, cos β, cos γ
By the definition of Direction Cosines
Question 19
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The signs of the X,Y and Z coordinates of a point that lies in the octant OXYZ ’is

A
(- , + , +)

B
(-, –, +)

C
( + , –, -)

D
( + , + , -)

Solution

X,Y,Z imply positive X,Y,Z axis &X ’,Y ’,Z ’imply negative X,Y,Z axis.
So, OXYZ ’will have a point of signs (+, +, -).
Question 20
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If a line in the ZX-plane makes an angle 30o with Z-axis, the direction cosines of this line are:

A

B

C

D

Solution

The line makes 30o with z-axis

Since it is in z-x axis the angle made with y-axis is 90

And angle made by x-axis is 60

Therefore direction cosines are

1/2, 0, √3/2