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: Understanding the Question:
We need to find the angle between two straight lines in 3D space, whose direction cosines \(l, m, n\) are bounded by a pair of given equations.

Step 2: Key Formula or Approach:
We will solve the two simultaneous equations to extract two distinct sets of direction ratios \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\).
The angle \(\theta\) between the lines is then found using the cosine formula:
\[ \cos \theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}} \]

Step 3: Detailed Explanation:
The given equations are:
1) \(l + m + n = 0 \implies l = -(m + n)\)
2) \(l^2 = m^2 + n^2\)
Substitute the expression for \(l\) from the first equation into the second equation:
\[ (-(m + n))^2 = m^2 + n^2 \] Expand the left side:
\[ m^2 + n^2 + 2mn = m^2 + n^2 \] Cancel \(m^2 + n^2\) from both sides:
\[ 2mn = 0 \implies mn = 0 \] This gives two cases: either \(m = 0\) or \(n = 0\).

Case 1: If \(m = 0\), substitute back into \(l + m + n = 0\):
\[ l + 0 + n = 0 \implies l = -n \] The direction ratios \((l_1, m_1, n_1)\) can be written as \((-n, 0, n)\). Dividing by \(n\), we get the simplest ratios: \(\vec{d_1} = \langle -1, 0, 1 \rangle\).

Case 2: If \(n = 0\), substitute back into \(l + m + n = 0\):
\[ l + m + 0 = 0 \implies l = -m \] The direction ratios \((l_2, m_2, n_2)\) can be written as \((-m, m, 0)\). Dividing by \(m\), we get the simplest ratios: \(\vec{d_2} = \langle -1, 1, 0 \rangle\).
Now, calculate the angle \(\theta\) between the two vectors \(\vec{d_1}\) and \(\vec{d_2}\):
\[ \cos \theta = \frac{|(-1)(-1) + (0)(1) + (1)(0)|}{\sqrt{(-1)^2 + 0^2 + 1^2} \sqrt{(-1)^2 + 1^2 + 0^2}} \] \[ \cos \theta = \frac{|1 + 0 + 0|}{\sqrt{2} \cdot \sqrt{2}} \] \[ \cos \theta = \frac{1}{2} \] Since \(\cos \theta = 1/2\), the acute angle \(\theta\) is \(\pi/3\) (or \(60^\circ\)).

Step 4: Final Answer:
The correct choice is (A).