Question

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

$\begin{array}{c}\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0,\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=5,\left|\overrightarrow{c}\right|=7\\ Letanglebetween\overrightarrow{a}and\overrightarrow{b}is\theta \\ \overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}\\ squaringonbothside\\ \Rightarrow {\left(\overrightarrow{a}+\overrightarrow{b}\right)}^2={\left(\overrightarrow{-c}\right)}^2\\ \Rightarrow {\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+2\overrightarrow{a}-\overrightarrow{b}={\left|\overrightarrow{c}\right|}^2\\ thenanglebetween\overrightarrow{a}and\overrightarrow{b}\\ \Rightarrow 3^2+5^2+2\left|a\right|\left|b\right|\cos \theta =72\\ \Rightarrow 9+25+2\times 3\times 5\cos \theta =49\\ \Rightarrow 34+30\cos \theta =49\\ \Rightarrow 30\cos \theta =49-34\\ \Rightarrow 30\cos \theta =15\\ \Rightarrow \cos \theta =\frac{1}{2}\\ \therefore \theta =\frac{\pi }{3}\end{array}$