Question-8 The angle between the lines whose direction cosines satisfy the equations l+m+n=0 and l^{2}=m^{2}+n^{2} is
A)
\frac\pi6B)
\frac\pi2C)
\frac\pi3D)
\frac\pi4Answer: C)
Explanation
l + m + n =0
m + n = -l
(m + n)2 = l2
l2 = m2 + n2 + 2mn
But, m2 + n2 = l2
2mn = 0
m = 0 or n = 0
l = —n or l = —m
l2 + m2 + n2 = 1
\therefore The two direction cosines are \left(\frac{1}{\sqrt{2}}, 0, \frac{-1}{\sqrt{2}}\right) and \left(\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}, 0\right)
\theta is the angle between them.
\therefore \cos \theta=l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}=\frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}}+0+0=\frac{1}{2}\therefore \theta=\frac{\pi}{3}.
Note : Taking l=\frac{-1}{\sqrt{2}}, will also given the same solution, but the supplementary angle. Both are correct, we choose the one present