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QUESTION: 1

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

Solution:

QUESTION: 2

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

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

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:

Solution:

QUESTION: 4

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

Solution:

QUESTION: 5

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

Solution:

QUESTION: 6

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

Solution:

QUESTION: 7

Find the direction cosines of the x axis.

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

The direction cosines of the line joining the points (2, -1, 8) and (-4, -3, 5) are:

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/(6^{2}+2^{2}+3^{2})1/2 , -2/(6^{2}+2^{2}+3^{2})^{1/2} , -3/(6^{2}+2^{2}+3^{2})^{1/2})

: ( -6/7, -2/7, -3/7)

QUESTION: 9

What are direction numbers of a line.

Solution:

The numbers which are proportional to direction cosines of a line are called direction numbers of the line.

QUESTION: 10

What are direction ratios of a line.

Solution:

QUESTION: 11

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)

Solution:

QUESTION: 12

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

Solution:

QUESTION: 13

If a line has the direction ratios -4, 18, -12 then what are its direction cosines?

Solution:

DR of the line : (-4, 18 -12)

DC of the line : (-4/k, 18/k, -12/k)

where k = ((4^{2}) + (18^{2}) + (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

The direction cosines of the line equally inclined with the axes, are:

Solution:
Cos^2 alpha+cos ^2 beta +cos^ 2 gamma =1

put alpha =beta. =gamma

we get cosalpha= 1/âˆš3

DC's of given line (1/âˆš3,1/âˆš3,1/3)

put alpha =beta. =gamma

we get cosalpha= 1/âˆš3

DC's of given line (1/âˆš3,1/âˆš3,1/3)

QUESTION: 15

If a line makes angles 45°,150°, 135°, with x, y and z-axes respectively, find its direction cosines.

Solution:

QUESTION: 16

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

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}

x^{2} +1 -2x + y^{2} + 4 - 4y + z^{2} + 9 – 6z = x^{2} +9 -6x + y^{2} + 4 - 4y + z^{2} + 1 + 2z

4x – 8z = 0

x – 2z = 0

QUESTION: 17

If a line in the ZX-plane makes an angle 60o with Z-axis, the direction cosines of this line are:

Solution:

QUESTION: 18

A line makes angles α, β, γ with the positive directions of X-axis, Y-axis and Z-axis, respectively, then the directions cosines of the line are:

Solution:

cos α, cos β, cos γ

By the definition of Direction Cosines

QUESTION: 19

The signs of the X,Y and Z coordinates of a point that lies in the octant OXYZ’ is

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

If a line in the ZX-plane makes an angle 30o with Z-axis, the direction cosines of this line are:

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, âˆš3/2

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, âˆš3/2

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