Question

The angle between the lines whose direction cosines satisfy the equations \(l + m + n = 0\) and \(l^2 = m^2 + n^2\) is:

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

Hint:

When dealing with a linear and a quadratic relationship involving direction cosines, always express one variable in terms of the other two using the linear equation and substitute it into the quadratic one to find the specific direction vectors.

Answer 1

The Correct Option is A

Step 1: Understand the setup.
The direction cosines $l, m, n$ of two lines satisfy both $l + m + n = 0$ and $l^2 = m^2 + n^2$. We want the angle between the two lines that result.
Step 2: Eliminate one variable.
From the first equation, $l = -(m + n)$. Substitute this into the second equation $l^2 = m^2 + n^2$.
Step 3: Simplify the result.
\[ (m+n)^2 = m^2 + n^2 \implies m^2 + 2mn + n^2 = m^2 + n^2 \] Cancelling gives $2mn = 0$, so either $m = 0$ or $n = 0$.
Step 4: Build the first direction.
Take $m = 0$. Then $l + n = 0$, so $l = -n$. A direction ratio set is $\vec{d_1} = \langle -1, 0, 1 \rangle$.
Step 5: Build the second direction.
Take $n = 0$. Then $l + m = 0$, so $l = -m$. A direction ratio set is $\vec{d_2} = \langle -1, 1, 0 \rangle$.
Step 6: Find the angle.
Using the cosine formula, \[ \cos\theta = \frac{|(-1)(-1) + (0)(1) + (1)(0)|}{\sqrt{2}\,\sqrt{2}} = \frac{1}{2} \] so $\theta = \frac{\pi}{3}$, which is option (A).
\[ \boxed{\theta = \dfrac{\pi}{3}} \]