Question

The angle between lines whose direction cosines satisfy the equation $l + m + n = 0$ and $l^2 - m^2 - n^2 = 0$, is ______.

• A. $\pi/2$
• B. $\pi/3$
• C. $\pi/4$
• D. $\pi/6$

Hint:

When dealing with homogeneous equations for direction cosines, directly substitute the linear equation into the quadratic one. It will always factor neatly into two roots, instantly giving you the direction vectors of the two lines!

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Solve the two equations simultaneously to find the two sets of direction cosines $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$.

Step 2: Formula Application:
Substitute $l = -(m+n)$ into the second equation. $(-(m+n))^2 - m^2 - n^2 = 0$.

Step 3: Explanation:
$(m^2 + n^2 + 2mn) - m^2 - n^2 = 0 \implies 2mn = 0$. Case 1: $m=0$. Then $l+n=0 \implies l = -n$. DRs are $(-n, 0, n) \to (-1, 0, 1)$. Case 2: $n=0$. Then $l+m=0 \implies l = -m$. DRs are $(-m, m, 0) \to (-1, 1, 0)$. $\cos \theta = \frac{|(-1)(-1) + (0)(1) + (1)(0)|}{\sqrt{2}\sqrt{2}} = \frac{1}{2}$. $\theta = \cos^{-1}(1/2) = \pi/3$.

Step 4: Final Answer:
The angle between the lines is $\pi/3$.