Question

The angle between the lines whose direction cosines are \((\frac{\sqrt3}{4},\frac{1}{4},\frac{\sqrt3}{2})\) and \((\frac{3}{4},\frac{1}{4},\frac{-\sqrt3}{2})\) is

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

Answer 1

The Correct Option is C

Let the direction cosines of the first line be \( (l_1, m_1, n_1) = \left( \frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2} \right) \). Let the direction cosines of the second line be \( (l_2, m_2, n_2) = \left( \frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2} \right) \).

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

Substitute the given direction cosines into the formula: \[ \cos \theta = \left( \frac{\sqrt{3}}{4} \right) \left( \frac{\sqrt{3}}{4} \right) + \left( \frac{1}{4} \right) \left( \frac{1}{4} \right) + \left( \frac{\sqrt{3}}{2} \right) \left( \frac{-\sqrt{3}}{2} \right) \] \[ \cos \theta = \frac{3}{16} + \frac{1}{16} - \frac{3}{4} \]

Simplify the expression: \[ \cos \theta = \frac{3 + 1}{16} - \frac{3 \times 4}{4 \times 4} \] \[ \cos \theta = \frac{4}{16} - \frac{12}{16} \] \[ \cos \theta = \frac{4 - 12}{16} \] \[ \cos \theta = \frac{-8}{16} \] \[ \cos \theta = -\frac{1}{2} \]

We need to find the angle \( \theta \) such that \( \cos \theta = -\frac{1}{2} \) and \( 0 \le \theta \le \pi \). We know that \( \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \). Since \( \cos \theta \) is negative, \( \theta \) lies in the second quadrant. Therefore, \( \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \).

The angle between two lines is conventionally taken as the acute angle (or \( \pi/2 \)). If the angle \( \theta \) obtained from the dot product formula is obtuse (\( \theta > \pi/2 \)), the acute angle between the lines is \( \pi - \theta \). In this case, \( \theta = \frac{2\pi}{3} \) is obtuse. The acute angle between the lines is \( \pi - \frac{2\pi}{3} = \frac{3\pi - 2\pi}{3} = \frac{\pi}{3} \).

The angle between the lines is (C): \( \frac{\pi}{3} \).