Question:
The angle between the lines whose direction cosines are \(\frac{-\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2}\) and \(\frac{-\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\) is
The angle between the lines whose direction cosines are \(\frac{-\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2}\) and \(\frac{-\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\) is
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Use cos of the angle equals the sum of products of the corresponding direction cosines.
Updated On: Nov 13, 2026
- \(90^{\circ}\)
- \(120^{\circ}\)
- \(45^{\circ}\)
- \(30^{\circ}\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept
If two lines have direction cosines \((l_1,m_1,n_1)\) and \((l_2,m_2,n_2)\), then \(\cos\theta=l_1l_2+m_1m_2+n_1n_2\).
Step 2: Calculation
\(\cos\theta=\left(\dfrac{-\sqrt3}{4}\right)\left(\dfrac{-\sqrt3}{4}\right)+\dfrac{1}{4}\cdot\dfrac{1}{4}+\left(\dfrac{-\sqrt3}{2}\right)\left(\dfrac{\sqrt3}{2}\right)\)
\(=\dfrac{3}{16}+\dfrac{1}{16}-\dfrac{3}{4}=-\dfrac{1}{2}\)
Step 3: Find the Angle
\(\cos\theta=-\dfrac{1}{2}\Rightarrow\theta=120^{\circ}\).
Step 4: Compare the options
\(120^{\circ}\) matches the second option.
Final Answer:
The angle between the direction vectors is 120 degrees. \[ \boxed{\text{(B) }120^{\circ}} \]
If two lines have direction cosines \((l_1,m_1,n_1)\) and \((l_2,m_2,n_2)\), then \(\cos\theta=l_1l_2+m_1m_2+n_1n_2\).
Step 2: Calculation
\(\cos\theta=\left(\dfrac{-\sqrt3}{4}\right)\left(\dfrac{-\sqrt3}{4}\right)+\dfrac{1}{4}\cdot\dfrac{1}{4}+\left(\dfrac{-\sqrt3}{2}\right)\left(\dfrac{\sqrt3}{2}\right)\)
\(=\dfrac{3}{16}+\dfrac{1}{16}-\dfrac{3}{4}=-\dfrac{1}{2}\)
Step 3: Find the Angle
\(\cos\theta=-\dfrac{1}{2}\Rightarrow\theta=120^{\circ}\).
Step 4: Compare the options
\(120^{\circ}\) matches the second option.
Final Answer:
The angle between the direction vectors is 120 degrees. \[ \boxed{\text{(B) }120^{\circ}} \]
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