Question

If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0,\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=5,\left|\overrightarrow{c}\right|=7$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

• A. $\pi /6$
• B. $2\pi /3$
• C. $5\pi /3$
• D. $\pi /3$

Answer 1

The correct option is D $\pi /3$

Given $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$
$\overrightarrow{c}=-(\overrightarrow{a}+\overrightarrow{b})$
$\Rightarrow$ ${\left|\overrightarrow{c}\right|}^2=\overrightarrow{c}\overrightarrow{c}=(\overrightarrow{a}+\overrightarrow{b}).(\overrightarrow{a}+\overrightarrow{b})$
$\Rightarrow {\left|\overrightarrow{c}\right|}^2={\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+2\overrightarrow{a}.\overrightarrow{b}$
$\Rightarrow {\left|\overrightarrow{c}\right|}^2={\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+2\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta$
$\Rightarrow 49=9+25+2\times 3\times 5\cos \theta$
$\Rightarrow 15=30\cos \theta \Rightarrow \cos \theta =\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3}$