Question

If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$, $\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=4,\left|\overrightarrow{c}\right|=5$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

• A. $\frac{\pi }{6}$
• B. $\frac{2\pi }{3}$
• C. $\frac{5\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (D)

$\frac{\pi }{2}$

$\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$ or $\overrightarrow{a}+\overrightarrow{b}=-\overrightarrow{c}$

Therefore

$|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{c}|$

$\sqrt{a^2+b^2+2ab\cos \theta }=c$

$a^2+b^2+2ab\cos \theta =c^2$

$3^2+4^2+2(3\times 4)\cos \theta =5^2$

$25+24\cos \theta =25$

$24\cos \theta =0$

Or

$\cos \theta =0$

Or

$\theta ={90}^0$.

Hence the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ${90}^0$.