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: Formula for angle between two lines.
The angle \( \theta \) between two lines is given by the formula: \[ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2, \] where \( (l_1, m_1, n_1) \) and \( (l_2, m_2, n_2) \) are the direction cosines of the two lines.

Step 2: Using the direction cosines.
Substitute the given direction cosines:
\[ \cos \theta = \left(\frac{\sqrt{3}}{4}\right)\left(\frac{-\sqrt{3}}{4}\right) + \left(\frac{1}{4}\right)\left(\frac{1}{2}\right) + \left(\frac{-\sqrt{3}}{2}\right)\left(\frac{1}{\sqrt{3}}\right). \]

Step 3: Simplifying the expression.
Simplify each term: \[ \cos \theta = -\frac{3}{16} + \frac{1}{8} - \frac{1}{2}. \]
Combine the terms: \[ \cos \theta = -\frac{3}{16} + \frac{2}{16} - \frac{8}{16} = -\frac{9}{16}. \]

Step 4: Finding the angle.
Now, find the angle \( \theta \) using the inverse cosine:
\[ \theta = \cos^{-1} \left( -\frac{9}{16} \right). \]
This gives: \[ \theta = 120^\circ. \] Final Answer:
The angle between the lines is:
\[ \boxed{120^\circ}. \]