Question

Let $\overline{a},\overline{b}$ and $\overline{c}$ be three vectors such that $\overline{a}+\overline{b}+\overline{c}=0$and $\left|\overline{a}\right|=3,\left|\overline{b}\right|=5,\left|\overline{c}\right|=7.$ Then an angle between $\overline{a}$ and $\overline{b}$ is

• A. ${15}^{\circ }$
• B. ${\cos }^{-1}\frac{2}{3}$
• C. .${30}^{\circ }$
• D. ${60}^{\circ }$

Answer 1

The correct option is D ${60}^{\circ }$

Given $\overline{a}+\overline{b}+\overline{c}=0$ and $|\overline{a}|=3,|\overline{b}|=5,|\overline{c}|=7$
Let angle between $\overline{a}$ and $\overline{b}$ be $\theta$
$\overline{a}+\overline{b}+\overline{c}=0$
$\Rightarrow \overline{a}+\overline{b}=-\overline{c}$
$\Rightarrow |\overline{a}{|}^2+|\overline{b}{|}^2+2\overline{a}\cdot \overline{b}=|\overline{c}{|}^2$
$\Rightarrow 9+25+30\cos \theta =49$
$\Rightarrow \cos \theta =\frac{1}{2}$
$\therefore \theta =\frac{\pi }{3}$
Hence, option D.