Question

If the relation between the direction ratios of two lines in \(\mathbb{R}^3\) are given by \(l + m + n = 0\), \(2lm + 2mn - ln = 0\), then the angle between the lines is:

• A. \(\frac{\pi}{6}\)
• B. \(\frac{2\pi}{3}\)
• C. \(\frac{\pi}{2}\)
• D. \(\frac{\pi}{4}\)

Hint:

The angle between two lines can be found using the dot product of their direction ratios. First, solve for the relationship between direction ratios and then apply the dot product formula.

Answer 1

The Correct Option is B

Step 1: We are given the direction ratios of two lines in 3D space, which are \((l_1, m_1, n_1)\) for the first line and \((l_2, m_2, n_2)\) for the second line.

Step 2: The formula to find the angle \(\theta\) between two lines with direction ratios \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\) is:

\[ \cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \cdot \sqrt{l_2^2 + m_2^2 + n_2^2}} \]

Step 3: We are given the relation between the direction ratios:

\(l + m + n = 0\)

\(2lm + 2mn - ln = 0\)

These are two equations in terms of the direction ratios of the lines. We can substitute these relationships into the formula for the cosine of the angle between the lines.

Step 4: After substituting the given conditions and solving for the cosine of the angle, we find that the angle between the lines is:

\[ \frac{2\pi}{3}. \]