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: The direction ratios of two lines in 3D space are provided: \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\).

Step 2: The angle \(\theta\) between the lines is calculated using:

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

Step 3: The direction ratios satisfy these relationships:

\(l + m + n = 0\)

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

These equations are used to simplify the angle calculation.

Step 4: After simplification, the angle between the lines is:

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