Question

If the direction ratio of two lines are given by and $l + 2 m + 3 n = 0$ , then the angle between the lines is

• A. $\frac { \pi } { 2 }$
• B. $\frac { \pi } { 3 }$
• C. $\frac { \pi } { 4 }$
• D. $\frac { \pi } { 6 }$

Answer 1

Answer: (A)

$\frac { \pi } { 2 }$

Answer 2

We have, $l + 2 m + 3 n = 0$ ……(i)

……(ii)

From equation (i), $l = - ( 2 m + 3 n )$

Putting the value of l in equation (ii)

⇒ $3 ( - 2 m - 3 n ) m + m n - 4 ( - 2 m - 3 n ) n = 0$

⇒ $- 6 m ^ { 2 } - 9 m n + m n + 8 m n + 12 n ^ { 2 } = 0$ ⇒ $6 m ^ { 2 } - 12 n ^ { 2 } = 0$

⇒ $m ^ { 2 } - 2 n ^ { 2 } = 0$

⇒ $m + \sqrt { 2 } n = 0$ or $m - \sqrt { 2 } n = 0$

$l + 2 m + 3 n = 0$ ……(i)

$0 . l + m + \sqrt { 2 } n = 0$ ……(iii)

$0 . l + m - \sqrt { 2 } n = 0$ ……(iv)

From equation (i) and equation (iii), $\frac { l } { 2 \sqrt { 2 } - 3 } = \frac { m } { - \sqrt { 2 } } = \frac { n } { 1 }$

From equation (i) and equation (iv), $\frac { l } { - 2 \sqrt { 2 } - 3 } = \frac { m } { \sqrt { 2 } } = \frac { n } { 1 }$

Thus, the direction ratios of two lines are $2 \sqrt { 2 } - 3 , - \sqrt { 2 } , 1$ and $- 2 \sqrt { 2 } - 3 , \sqrt { 2 } , 1$

$\left( l _ { 1 } , m _ { 1 } , n _ { 1 } \right) = ( 2 \sqrt { 2 } - 3 , - \sqrt { 2 } , 1 )$ $\left( l _ { 2 } , m _ { 2 } , n _ { 2 } \right) = ( - 2 \sqrt { 2 } - 3 , \sqrt { 2 } , 1 )$,

$l _ { 1 } l _ { 2 } + m _ { 1 } m _ { 2 } + n _ { 1 } n _ { 2 } = 0$.Hence, the angle between them π/2.