The Cartesian equation of a line AB is \(\frac { 2x-1 }{ \sqrt { 3 } } =\frac { y+2 }{ 2 } \frac { z-3 }{ 3 } \). Find the direction cosines of a line parallel to AB.

1

(a)

\(\text { Line is } \frac{2\left(x-\frac{1}{2}\right)}{\sqrt{3}}=\frac{y+2}{2}=\frac{z-3}{3} \)
\(\Rightarrow \frac{x-\frac{1}{2}}{\sqrt{3}}=\frac{y+2}{4}=\frac{z-3}{6} \)
\(\text { DR's of line } A B \text { are } \sqrt{3}, 4 \text { and } 6 \text {. }\)
\(\text { DR's of line parallel to } A B \text { are } \sqrt{3}, 4 \text { and } 6 \text {. }\)
\(\text { Dividing by } \sqrt{3+16+36}=\sqrt{55} \text {, we get }\)
Direction cosines are \(\frac { \sqrt { 3 } }{ \sqrt { 55 } } ,\frac { 4 }{ \sqrt { 55 } } ,\frac { 6 }{ \sqrt { 55 } } \).

Find the direction cosines of the line passing through the following points: (-2, 4, -5), (1, 2, 3).

1

(a)

DR's are I + 2, 2 - 4, 3 + 5, i.e 3, - 2, 8.
Divide by \(\sqrt { 9+4+64 } =\sqrt { 77 } \);
DC's are \(\frac { 3 }{ \sqrt { 77 } } ,\frac { -2 }{ \sqrt { 77 } } ,\frac { 8 }{ \sqrt { 77 } } \)

Write the Cartesian equation of the following line given in vector form: \(\overrightarrow { r } =2\hat { i } +\hat { j } +4\hat { k } +\lambda (\hat { i } +\hat { j } -\hat { k } )\)

1

(a)

Point through which line passes is (2, I, - 4) and dr's: 1, 1,-1.
.'. Cartesian equation of line is \(\frac { x-2 }{ 1 } =\frac { y-1 }{ -1 } =\frac { z+4 }{ -1 } \)

What are the direction cosines of a line, which makes equal angles with the coordinate axes?

1

(a)

\(\alpha =\beta=\gamma\)
\(\Rightarrow 3 \cos ^{2} \alpha =1\)
DC's: \(\pm \frac { 1 }{ \sqrt { 3 } } \pm \frac { 1 }{ \sqrt { 3 } } \pm \frac { 1 }{ \sqrt { 3 } } \)

If a line has a direction ratios 2, -1, -2, then what are its direction cosines?

1

(a)

Dr's are 2,-1,-2;Divide by \(\sqrt { 4+1+4 } =3\). ;DC's are \(\frac { 2 }{ 3 } ,-\frac { 1 }{ 3 } ,-\frac { 2 }{ 3 } \)

Find the distance of the plane 3x - 4y + 12z = 3 from the origin.

1

(a)

Distance = \(\left| \frac { 0-0+0-3 }{ \sqrt { 9+16+144 } } \right| =\frac { 3 }{ 13 } \) units 

Write the direction cosines of a line parallel to the z-axis.

1

(a)

DC's are 0,0,1