Question

If P(0, 1, 2), Q(4, –2, 1) and O(0, 0, 0) are three points, then ∠POQ =

(a) $\frac{\mathrm{\pi }}{6}$

(b) $\frac{\mathrm{\pi }}{4}$

(c) $\frac{\mathrm{\pi }}{3}$

(d) $\frac{\mathrm{\pi }}{2}$

Answer 1

(d) $\frac{\mathrm{\pi }}{2}$

$$\begin{gathered} P{Q}^2={\left(4-0\right)}^2+{\left(-2-1\right)}^2+{\left(1-2\right)}^2=16+9+1=26 \\ O{P}^2={\left(0-0\right)}^2+{\left(1-0\right)}^2+{\left(2-0\right)}^2=0+1+4=5 \\ Q{O}^2={\left(0-4\right)}^2+{\left(0+2\right)}^2+{\left(0-1\right)}^2=16+1+4=21 \\ \mathrm{Since},P{Q}^2=O{P}^2+Q{O}^2 \\ \mathrm{Hence},\angle POQ=\frac{\mathrm{\pi }}{2} \end{gathered}$$