Question

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

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{6}$

Answer 1

Answer: (A)

$\vec{a}+\vec{b}+\vec{c}=\vec{0}$
$\Rightarrow \vec{a}+\vec{b}=-\vec{c}$
$\Rightarrow {\left(\vec{a}+\vec{b}\right)}^2={\vec{c}}^2$
$\Rightarrow {\vec{a}}^2+{\vec{b}}^2+2\vec{a}.\vec{b}={\vec{c}}^2$
$\Rightarrow 9+25+2\left(3\right)(5)cos\theta =49$
$\therefore cos\theta =\frac{49-34}{30}=\frac{1}{2}$
$\therefore \theta =60^{\circ }=\frac{\pi }{3}$