A) \[\frac{\pi }{4}\]
B) \[\frac{\pi }{3}\]
C) \[\frac{2\pi }{3}\]
D) \[\frac{\pi }{6}\]
Correct Answer: B
Solution :
| If \[\vec{a}+\vec{b}+\vec{c}=0,\] \[|\vec{a}|=3,\] \[|\vec{b}|=5,\] \[|\vec{c}|=7\] |
| \[\Rightarrow \,\,\,\,\vec{a}+\vec{b}=-\,\vec{c}\,\Rightarrow \,\,{{\left( \vec{a}+\vec{b} \right)}^{2}}={{\vec{c}}^{2}}\]\[\Rightarrow \,\,\,{{\vec{a}}^{2}}+{{\vec{b}}^{2}}+2\vec{a}.\vec{b}\,\cos \theta ={{\vec{c}}^{2}}\] |
| \[\Rightarrow \,\,\,{{3}^{2}}+{{5}^{2}}+2\times 3\times 5\cos \theta ={{7}^{2}}\]\[\Rightarrow \,\,\,9+25+30\cos \theta =49\] |
| \[\Rightarrow \,\,\,30\cos \theta =49-34\]\[\Rightarrow \,\,\,30\cos \theta -15\Rightarrow cos\theta =\frac{15}{30}\] |
| \[\Rightarrow \,\,\,\,\cos \theta =\frac{1}{2}\Rightarrow \cos \theta =\cos \frac{\pi }{3}\]\[\Rightarrow \,\,\,\theta =\frac{\pi }{3}\] |
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