A) \[\frac{\pi }{6}\]
B) \[\frac{\pi }{4}\]
C) \[\frac{\pi }{3}\]
D) \[\frac{\pi }{2}\]
Correct Answer: C
Solution :
Given,\[|\overrightarrow{a}|=5,|\overrightarrow{b}|=3,|\overrightarrow{c}|=7\] and\[\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}\] \[\Rightarrow \] \[\overrightarrow{c}=-(\overrightarrow{a}+\overrightarrow{b})\] \[\Rightarrow \] \[|\overrightarrow{c}|=|-(\overrightarrow{a}+\overrightarrow{b}){{|}^{2}}\] \[\Rightarrow \] \[|\overrightarrow{c}{{|}^{2}}=|\overrightarrow{a}{{|}^{2}}+|\overrightarrow{b}{{|}^{2}}+2|\overrightarrow{a}||\overrightarrow{b}|\cos \theta \] \[\Rightarrow \] \[{{(7)}^{2}}={{(5)}^{2}}+{{(3)}^{2}}+2(5)(3)\cos \theta \] \[\Rightarrow \] \[49+25+9+30\cos \theta \] \[\Rightarrow \] \[30\cos \theta =15\] \[\Rightarrow \] \[\cos \theta =\frac{15}{30}=\frac{1}{2}=\cos \frac{\pi }{3}\] \[\Rightarrow \] \[\theta =\frac{\pi }{3}\]
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