Question

Let \( P(3, 2, 3) \), \( Q(4, 6, 2) \), and \( R(7, 3, 2) \) be the vertices of \( \triangle PQR \). Then, the angle \( \angle QPR \) is

• A. \( \frac{\pi}{6} \)
• B. \( \cos^{-1} \left( \frac{7}{18} \right) \)
• C. \( \cos^{-1} \left( \frac{1}{18} \right) \)
• D. \( \frac{\pi}{3} \)

Answer 1

The Correct Option is D

Solution: To find the angle ∠QPR, we calculate the direction ratios of PR and PQ.

Step 1. Direction Ratio of PR:  
  PR = (7 − 3, 3 − 2, 2 − 3) = (4, 1, −1)

Step 2. Direction Ratio of PQ:
  PQ = (4 − 3, 6 − 2, 2 − 3) = (1, 4, −1)

Step 3. Calculating cosθ:
  \(\cosθ = \frac{4·1 + 1·4 + (−1)·(−1)}{\sqrt{18}·\sqrt{18}} = \frac{4 + 4 + 1}{18} = \frac{9}{18} = \frac{1}{2}\)
Step 4. Therefore:

  \(θ = \cos⁻¹\left(\frac{1}{2}\right) = \frac{π}{3}\)

The Correct Answer is:\( \frac{π}{3} \)