A) \[0\]
B) \[-1\]
C) \[-\frac{3}{2}\]
D) \[3\]
Correct Answer: C
Solution :
Given that, \[|\overset{\to }{\mathop{\mathbf{a}}}\,|\,\,=\,\,|\overset{\to }{\mathop{\mathbf{b}}}\,|\,\,=\,\,|\overset{\to }{\mathop{\mathbf{c}}}\,|\,\,=1\] and \[\overset{\to }{\mathop{\mathbf{a}}}\,+\overset{\to }{\mathop{\mathbf{b}}}\,+\overset{\to }{\mathop{\mathbf{c}}}\,=0\] \[\Rightarrow \] \[{{(\overset{\to }{\mathop{\mathbf{a}}}\,+\overset{\to }{\mathop{\mathbf{b}}}\,+\overset{\to }{\mathop{\mathbf{c}}}\,)}^{2}}=0\] \[\Rightarrow \] \[|\overset{\to }{\mathop{\mathbf{a}}}\,{{|}^{2}}+|\overset{\to }{\mathop{\mathbf{b}}}\,{{|}^{2}}+|\overset{\to }{\mathop{\mathbf{c}}}\,{{|}^{2}}\] \[+2(\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,+\overset{\to }{\mathop{\mathbf{b}}}\,\cdot \overset{\to }{\mathop{\mathbf{c}}}\,+\overset{\to }{\mathop{\mathbf{c}}}\,\cdot \overset{\to }{\mathop{\mathbf{a}}}\,)=0\] \[\Rightarrow \] \[1+1+1+2(\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,+\overset{\to }{\mathop{\mathbf{b}}}\,\cdot \overset{\to }{\mathop{\mathbf{c}}}\,+\overset{\to }{\mathop{\mathbf{c}}}\,\cdot \overset{\to }{\mathop{\mathbf{a}}}\,)=0\] \[\Rightarrow \] \[\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,+\overset{\to }{\mathop{\mathbf{b}}}\,\cdot \overset{\to }{\mathop{\mathbf{c}}}\,+\overset{\to }{\mathop{\mathbf{c}}}\,\cdot \overset{\to }{\mathop{\mathbf{a}}}\,=-\frac{3}{2}\]
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