Three-dimensional Geometry Test

Result

Three-dimensional Geometry Test - 5

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Weekly Quiz Competition

Question 1
1 / -0.25

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 2
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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
Question 3
1 / -0.25

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 4
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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 5
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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 6
1 / -0.25

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 7
1 / -0.25

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 8
1 / -0.25

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 9
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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 10
1 / -0.25

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