A) \[1\]
B) \[2\]
C) \[0\]
D) None of these
Correct Answer: C
Solution :
Key Idea: If \[\overset{\to }{\mathop{\mathbf{a}}}\,,\,\,\overset{\to }{\mathop{\mathbf{b}}}\,\] and \[\overset{\to }{\mathop{\mathbf{c}}}\,\] are non-coplanar vectors, then\[[\overset{\to }{\mathop{\mathbf{a}}}\,\overset{\to }{\mathop{\mathbf{b}}}\,\overset{\to }{\mathop{\mathbf{c}}}\,]\ne 0\]. \[\therefore \] \[\frac{\overset{\to }{\mathop{\mathbf{a}}}\,\cdot (\overset{\to }{\mathop{\mathbf{b}}}\,\times \overset{\to }{\mathop{\mathbf{c}}}\,)}{(\overset{\to }{\mathop{\mathbf{c}}}\,\times \overset{\to }{\mathop{\mathbf{a}}}\,)\cdot \overset{\to }{\mathop{\mathbf{b}}}\,}+\frac{\overset{\to }{\mathop{\mathbf{b}}}\,\cdot (\overset{\to }{\mathop{\mathbf{a}}}\,\times \overset{\to }{\mathop{\mathbf{c}}}\,)}{\overset{\to }{\mathop{\mathbf{c}}}\,\cdot (\overset{\to }{\mathop{\mathbf{a}}}\,\times \overset{\to }{\mathop{\mathbf{b}}}\,)}\] \[=\frac{[\overset{\to }{\mathop{\mathbf{a}}}\,\overset{\to }{\mathop{\mathbf{b}}}\,\overset{\to }{\mathop{\mathbf{c}}}\,]}{[\overset{\to }{\mathop{\mathbf{c}}}\,\overset{\to }{\mathop{\mathbf{b}}}\,\overset{\to }{\mathop{\mathbf{a}}}\,]}+\frac{[\overset{\to }{\mathop{\mathbf{b}}}\,\overset{\to }{\mathop{\mathbf{a}}}\,\overset{\to }{\mathop{\mathbf{c}}}\,]}{[\overset{\to }{\mathop{\mathbf{c}}}\,\overset{\to }{\mathop{\mathbf{a}}}\,\overset{\to }{\mathop{\mathbf{b}}}\,]}\] \[=\frac{[\overset{\to }{\mathop{\mathbf{a}}}\,\overset{\to }{\mathop{\mathbf{b}}}\,\overset{\to }{\mathop{\mathbf{c}}}\,]-[\overset{\to }{\mathop{\mathbf{a}}}\,\overset{\to }{\mathop{\mathbf{b}}}\,\overset{\to }{\mathop{\mathbf{c}}}\,]}{[\overset{\to }{\mathop{\mathbf{c}}}\,\overset{\to }{\mathop{\mathbf{a}}}\,\overset{\to }{\mathop{\mathbf{b}}}\,]}=0\]
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