# What is the angle between two lines whose direction ratios satisfy following equations?

## Find the angle between two lines whose direction ratios satisfy $2 l - m + 2 n = 0$ and $m n + n l + l m = 0$

Mar 12, 2017

The two lines are perpendicular to each other.

#### Explanation:

Let the direction cosines of the two lines be $\left({l}_{1} , {m}_{1} , {n}_{1}\right)$ and $\left({l}_{2} , {m}_{2} , {n}_{2}\right)$ and as they satisfy the conditions we have

$2 l - m + 2 n = 0$ .............................(1) and

$m n + n l + l m = 0$ .............................(2)

Note that if $\left({a}_{1} , {b}_{1} , {c}_{1}\right)$ and $\left({a}_{2} , {b}_{2} , {c}_{2}\right)$ are direction ratios of two lines, we have ${l}_{1} / {a}_{1} = {m}_{1} / {b}_{1} = {n}_{1} / {c}_{1}$ and ${l}_{2} / {a}_{2} = {m}_{2} / {b}_{2} = {n}_{2} / {c}_{2}$

and the angle $\theta$ between them is given by

$\cos \theta = \frac{{a}_{1} {a}_{2} + {b}_{1} {b}_{2} + {c}_{1} {c}_{2}}{\sqrt{{a}_{1}^{2} + {b}_{1}^{2} + {c}_{1}^{2}} \sqrt{{a}_{2}^{2} + {b}_{2}^{2} + {c}_{2}^{2}}}$

From (1), we get $m = 2 \left(l + n\right)$ and substituting in (2) we get

$2 n \left(l + n\right) + n l + 2 l \left(l + n\right) = 0$ or $2 {l}^{2} + 5 \ln + 2 {n}^{2} = 0$

or $\left(l + 2 n\right) \left(2 l + n\right) = 0$

i.e either $l + 2 n = 0$ or $2 l + n = 0$

if $l + 2 n = 0$, $l = - 2 n$ and $m = 2 \left(- 2 n + n\right) = - 2 n$ and we have

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

and if $2 l + n = 0$m $n = - 2 l$ and $m = 2 \left(l - 2 l\right) = - 2 l$ and we have

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

and $\cos \theta = \frac{{a}_{1} {a}_{2} + {b}_{1} {b}_{2} + {c}_{1} {c}_{2}}{\sqrt{{a}_{1}^{2} + {b}_{1}^{2} + {c}_{1}^{2}} \sqrt{{a}_{2}^{2} + {b}_{2}^{2} + {c}_{2}^{2}}}$

= $\frac{\left(- 2\right) \times 1 + \left(- 2\right) \times \left(- 2\right) + 1 \times \left(- 2\right)}{\sqrt{4 + 4 + 1} \sqrt{1 + 4 + 4}}$

= $\frac{- 2 + 4 - 2}{9} = 0$

Hence the two lines are perpendicular to each other.