Question

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

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

Answer 1

The correct option is D $\frac{\pi }{3}$

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