Question

The angle between the lines whose direction cosines are \[ \frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2} \quad \text{and} \quad \frac{-\sqrt{3}}{4}, \frac{1}{2}, \frac{1}{\sqrt{3}} \] is

• A. \( 90^\circ \)
• B. \( 120^\circ \)
• C. \( 45^\circ \)
• D. \( 30^\circ \)

Hint:

To find the angle between two lines, use the formula for the cosine of the angle involving the direction cosines of the lines.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The angle \( \theta \) between two lines with direction cosines \( (l_1, m_1, n_1) \) and \( (l_2, m_2, n_2) \) is given by \( \cos \theta = l_1l_2 + m_1m_2 + n_1n_2 \).
Step 2: Detailed Explanation:
Direction Cosines:
\( L_1 = (-\sqrt{3}/4, 1/4, -\sqrt{3}/2) \)
\( L_2 = (-\sqrt{3}/4, 1/4, \sqrt{3}/2) \)
\( \cos \theta = \left(\frac{-\sqrt{3}}{4}\right)\left(\frac{-\sqrt{3}}{4}\right) + \left(\frac{1}{4}\right)\left(\frac{1}{4}\right) + \left(\frac{-\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) \)
\( \cos \theta = \frac{3}{16} + \frac{1}{16} - \frac{3}{4} \)
\( \cos \theta = \frac{4}{16} - \frac{12}{16} = -\frac{8}{16} = -\frac{1}{2} \).
Since \( \cos \theta = -1/2 \), \( \theta = 120^\circ \).
Step 3: Final Answer:
The angle is \( 120^\circ \).