<p>The angle between the lines whose direction cosines satisfy the equations \(l + m + n = 0\) and \(l^2 = m^2 + n^2\) is</p>
<p>(a) \(\frac{\pi}{3}\)</p>
<p>(b) \(\frac{\pi}{4}\)</p>
<p>(c) \(\frac{\pi}{6}\)</p>
<p>(d) \(\frac{\pi}{2}\)</p>
Step-by-Step Solution
Key Concept: Use the constraint equations to find the direction cosines, then apply the angle formula between two lines using the dot product of direction cosines.
Step 1: Given $l + m + n = 0$, so $l = -(m + n)$ Step 2: Squaring both sides: $(m + n)^2 = l^2$, which gives $m^2 + n^2 + 2mn = m^2 + n^2$ Step 3: Since $l^2 = m^2 + n^2$ (given), we have $2mn = 0$ Step 4 (Case I): When $m = 0$, then $l = -n$. So direction cosines are $(l, m, n) = (1, 0, -1)$ (after normalization) Step 5 (Case II): When $n = 0$, then $l = -m$. So direction cosines are $(l, m, n) = (1, -1, 0)$ (after normalization) Step 6: Using $\cos\theta = |l_1 l_2 + m_1 m_2 + n_1 n_2|$ $\cos\theta = |(1)(1) + (0)(-1) + (-1)(0)| = |1| = \frac{1}{2}$ Step 7: Therefore $\theta = \frac{\pi}{3}$ ∴ Answer is (a)
Correct Answer: A