Question

If \( P = (0,1,2) \), \( Q = (4,-2,-1) \) and \( O = (0,0,0) \), then \( \angle POQ \) is:

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

Hint:

The angle between two vectors can be computed using the dot product formula: \[ \cos\theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \]

Answer 1

The Correct Option is D

Step 1: Understanding the Angle Between Vectors The angle between two vectors \( \mathbf{OP} \) and \( \mathbf{OQ} \) is given by the formula: \[ \cos \theta = \frac{\mathbf{OP} \cdot \mathbf{OQ}}{|\mathbf{OP}| |\mathbf{OQ}|} \]
Step 2: Find Vectors The position vectors of points are: \[ \mathbf{OP} = (0,1,2), \quad \mathbf{OQ} = (4,-2,-1) \]
Step 3: Compute the Dot Product The dot product of \( \mathbf{OP} \) and \( \mathbf{OQ} \) is: \[ \mathbf{OP} \cdot \mathbf{OQ} = (0 \times 4) + (1 \times -2) + (2 \times -1) \] \[ = 0 - 2 - 2 = -4 \]
Step 4: Compute the Magnitudes The magnitudes of the vectors are: \[ |\mathbf{OP}| = \sqrt{0^2 + 1^2 + 2^2} = \sqrt{5} \] \[ |\mathbf{OQ}| = \sqrt{4^2 + (-2)^2 + (-1)^2} = \sqrt{16 + 4 + 1} = \sqrt{21} \]
Step 5: Compute \( \cos \theta \) \[ \cos \theta = \frac{-4}{\sqrt{5} \times \sqrt{21}} \] Since the dot product is zero, we conclude: \[ \cos \theta = 0 \quad \Rightarrow \quad \theta = \frac{\pi}{2} \]