Question

If $\theta$ is the angle between $\overrightarrow{p}=a\hat{i}+b\hat{j}+c\hat{k},\overrightarrow{q}=b\hat{i}+c\hat{j}+a\hat{k}$, then $\theta$ is in

• A. $[0,\frac{\pi }{2}]$
• B. $[\frac{\pi }{2},\pi ]$
• C. $[\frac{\pi }{3}\frac{2\pi }{3}]$
• D. $[0,\frac{2\pi }{3}]$

Answer 1

The correct option is D $[0,\frac{2\pi }{3}]$

$\cos \theta =\frac{ab+bc+ca}{a^2+b^2+c^2}=\frac{1}{2}\frac{{(a+b+c)}^2-(a^2+b^2+c^2)}{a^2+b^2+c^2}=\frac{1}{2}[\frac{{(a+b+c)}^2}{a^2+b^2+c^2}-1]$
.
From the above expression, $\cos \theta$ has minimum value when $a+b+c=0$, , which gives $\cos {\theta }_{min}=-\frac{1}{2}$
.
Similarly when $a=b=c$ we get $\frac{{(a+b+c)}^2}{a^2+b^2+c^2}=3$, which gives the maximum value of $\cos \theta$ as $\cos {\theta }_{max.}=1$
Hence $\cos \theta \in [-\frac{1}{2},1]$
$\Rightarrow \theta \in [0,\frac{2\pi }{3}]$