Let P(3, 2, 3), Q(4, 6, 2) and R(7, 3, 2) be the vertices of . Then the angle is [2024]

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Solution

Q.
Let P(3, 2, 3), Q(4, 6, 2) and R(7, 3, 2) be the vertices of $△ P Q R$ . Then the angle $∠ Q P R$ is [2024]

1 $\cos^{- 1} (\frac{7}{18})$

2 $\cos^{- 1} (\frac{1}{18})$

3 $\frac{π}{3}$

4 $\frac{π}{6}$

Ans.
(3)

Direction ratios of line PQ = <4 – 3, 6 – 2, 2 – 3> = <1, 4, –1>

Direction ratios of line PR = <7 – 3, 3 – 2, 2 – 3> = <4, 1, –1>

Let $θ$ be the required angle.

$∴ \cos θ =$ $| \frac{1 \times 4 + 4 \times 1 + (- 1) \times (- 1)}{\sqrt{1 + 16 + 1} \sqrt{16 + 1 + 1}} |$ $=$ $| \frac{9}{18} |$ $= \frac{1}{2}$

$\Rightarrow θ = \cos^{- 1} (\frac{1}{2}) = \frac{π}{3} ∴ ∠ Q P R = \frac{π}{3}$ .

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