The angle between the lines whose direction cosine satisfy the...

We have,&

$l + m + n = 0$ & $. . . ( i )$

and $l^{2} = m^{2} + n^{2}$ & $. . . ( ii )$

Putting the value of $l$ in Eq. $( ii )$, we get

$( m + n{)}^{2} = m^{2} + n^{2}$

$⇒ m^{2} + n^{2} + 2mn = m^{2} + n^{2}$

$⇒ mn = 0$

$⇒ m = 0, n = 0$

when $m = 0, l = - n$

$∴$ Direction cosines are \left(- \dfrac{1}{2}, 0, \dfrac{1}{2}\right)$.</p><p>When$n = 0, l = - m$</p><p>$∴$Direction cosines are \left(- \dfrac{1}{2}, \dfrac{1}{2}, 0\right)$.

Angle between lines are

$cos\theta = \left(\dfrac{- 1}{2} \times - \dfrac{1}{2}\right) + \left(0 \times \dfrac{1}{2}\right) +\left(\dfrac{1}{2} \times0\right)$

cos\theta = $\dfrac{1}{2}, \theta = \pi$$/ 3$