Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be three vectors such that they are pairwise inclined at the same angle $\theta$ and $\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=3,\left|\overrightarrow{c}\right|=4$, then $\theta$ belongs to

• A. $[0,\frac{\pi }{2}]$
• B. $[0,{\cos }^{-1}\frac{29}{52}]$
• C. $[0,\pi -{\cos }^{-1}\frac{29}{52}]$
• D. None of these

Answer 1

Answer: (C)

$[0,\pi -{\cos }^{-1}\frac{29}{52}]$

$|\overline{a}+\overline{b}+\overline{c}{|}^2\ge 0$
i.e.,${\left|\overline{a}\right|}^2+{\left|\overline{b}\right|}^2{|+\overline{c}|}^2+2[\overline{a}.\overline{b}+\overline{b}.\overline{c}+\overline{c}.\overline{a}]\ge 0$
29$+2\cos \theta [6+12+8]\ge 0$
$29+52\cos \theta \ge 0$
$\cos \theta \ge -\frac{29}{52}$
$\Rightarrow \cos \theta \ge \pi -\frac{29}{52}$

Hence, option '$C$' is correct.