Question

The angle between the lines, whose direction cosines are proportional to 4,√3-1,-√3-1 and 4,-√3-1,√3-1, is

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

Answer 1

The Correct Option is C

We are given two lines with direction cosines proportional to:

Line 1: \( \langle 4, \sqrt{3} - 1, -\sqrt{3} - 1 \rangle \)

Line 2: \( \langle 4, -\sqrt{3} - 1, \sqrt{3} - 1 \rangle \)

To find the angle \( \theta \) between these lines, we use the dot product formula for direction ratios:

\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \]

First, compute the dot product of the direction ratios:

\[ 4 \cdot 4 + (\sqrt{3} - 1)(-\sqrt{3} - 1) + (-\sqrt{3} - 1)(\sqrt{3} - 1) \]

\[ = 16 + \left[ -(\sqrt{3})^2 - \sqrt{3} + \sqrt{3} + 1 \right] + \left[ -(\sqrt{3})^2 + \sqrt{3} - \sqrt{3} + 1 \right] \]

\[ = 16 + \left[ -3 + 1 \right] + \left[ -3 + 1 \right] \]

\[ = 16 - 2 - 2 = 12 \]

Next, compute the magnitudes of the direction ratios:

For Line 1:

\[ \sqrt{4^2 + (\sqrt{3} - 1)^2 + (-\sqrt{3} - 1)^2} \]

\[ = \sqrt{16 + (3 - 2\sqrt{3} + 1) + (3 + 2\sqrt{3} + 1)} \]

\[ = \sqrt{16 + 4 - 2\sqrt{3} + 4 + 2\sqrt{3}} \]

\[ = \sqrt{24} = 2\sqrt{6} \]

For Line 2:

\[ \sqrt{4^2 + (-\sqrt{3} - 1)^2 + (\sqrt{3} - 1)^2} \]

\[ = \sqrt{16 + (3 + 2\sqrt{3} + 1) + (3 - 2\sqrt{3} + 1)} \]

\[ = \sqrt{16 + 4 + 2\sqrt{3} + 4 - 2\sqrt{3}} \]

\[ = \sqrt{24} = 2\sqrt{6} \]

Now, substitute into the dot product formula:

\[ \cos \theta = \frac{12}{(2\sqrt{6})(2\sqrt{6})} = \frac{12}{24} = \frac{1}{2} \]

Thus, the angle \( \theta \) is:

\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

Answer 2

Step 1: Understand the problem and given information.

We are tasked with finding the angle between two lines whose direction cosines are proportional to:

• Line 1: \( 4, \sqrt{3} - 1, -\sqrt{3} - 1 \)
• Line 2: \( 4, -\sqrt{3} - 1, \sqrt{3} - 1 \)

The angle \( \theta \) between two lines can be found using the formula:

\[ \cos\theta = \frac{l_1l_2 + m_1m_2 + n_1n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \cdot \sqrt{l_2^2 + m_2^2 + n_2^2}}, \]

where \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \) are the direction ratios of the two lines.

Step 2: Assign direction ratios for the two lines.

Let the direction ratios for Line 1 be:

\[ l_1 = 4, \quad m_1 = \sqrt{3} - 1, \quad n_1 = -\sqrt{3} - 1. \]

Let the direction ratios for Line 2 be:

\[ l_2 = 4, \quad m_2 = -\sqrt{3} - 1, \quad n_2 = \sqrt{3} - 1. \]

Step 3: Compute the numerator of \( \cos\theta \).

The numerator is given by:

\[ l_1l_2 + m_1m_2 + n_1n_2. \]

Substitute the values:

\[ l_1l_2 = 4 \cdot 4 = 16, \]

\[ m_1m_2 = (\sqrt{3} - 1)(-\sqrt{3} - 1) = -(\sqrt{3} - 1)(\sqrt{3} + 1) = -(3 - 1) = -2, \]

\[ n_1n_2 = (-\sqrt{3} - 1)(\sqrt{3} - 1) = -(\sqrt{3} + 1)(\sqrt{3} - 1) = -(3 - 1) = -2. \]

Add these terms:

\[ l_1l_2 + m_1m_2 + n_1n_2 = 16 - 2 - 2 = 12. \]

Step 4: Compute the denominator of \( \cos\theta \).

The denominator is given by:

\[ \sqrt{l_1^2 + m_1^2 + n_1^2} \cdot \sqrt{l_2^2 + m_2^2 + n_2^2}. \]

First, compute \( l_1^2 + m_1^2 + n_1^2 \):

\[ l_1^2 = 4^2 = 16, \quad m_1^2 = (\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}, \]

\[ n_1^2 = (-\sqrt{3} - 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}. \]

Add these terms:

\[ l_1^2 + m_1^2 + n_1^2 = 16 + (4 - 2\sqrt{3}) + (4 + 2\sqrt{3}) = 16 + 4 + 4 = 24. \]

Thus:

\[ \sqrt{l_1^2 + m_1^2 + n_1^2} = \sqrt{24} = 2\sqrt{6}. \]

Next, compute \( l_2^2 + m_2^2 + n_2^2 \):

Since the direction ratios of Line 2 are symmetric to those of Line 1, we have:

\[ l_2^2 + m_2^2 + n_2^2 = 24. \]

Thus:

\[ \sqrt{l_2^2 + m_2^2 + n_2^2} = \sqrt{24} = 2\sqrt{6}. \]

The denominator becomes:

\[ \sqrt{l_1^2 + m_1^2 + n_1^2} \cdot \sqrt{l_2^2 + m_2^2 + n_2^2} = (2\sqrt{6})(2\sqrt{6}) = 24. \]

Step 5: Compute \( \cos\theta \).

Substitute into the formula for \( \cos\theta \):

\[ \cos\theta = \frac{l_1l_2 + m_1m_2 + n_1n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \cdot \sqrt{l_2^2 + m_2^2 + n_2^2}} = \frac{12}{24} = \frac{1}{2}. \]

Step 6: Find the angle \( \theta \).

If \( \cos\theta = \frac{1}{2} \), then:

\[ \theta = \frac{\pi}{3}. \]

Final Answer:

\( \frac{\pi}{3} \)