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)

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