A) \[[\overset{\to }{\mathop{\mathbf{A}}}\,\overset{\to }{\mathop{\mathbf{B}}}\,\overset{\to }{\mathop{\mathbf{C}}}\,]\]
B) \[[\overset{\to }{\mathop{\mathbf{B}}}\,\overset{\to }{\mathop{\mathbf{A}}}\,\overset{\to }{\mathop{\mathbf{C}}}\,]\]
C) \[0\]
D) \[1\]
Correct Answer: C
Solution :
\[\overset{\to }{\mathop{\mathbf{A}}}\,\cdot \{(\overset{\to }{\mathop{\mathbf{B}}}\,+\overset{\to }{\mathop{\mathbf{C}}}\,)\times (\overset{\to }{\mathop{\mathbf{A}}}\,+\overset{\to }{\mathop{\mathbf{B}}}\,+\overset{\to }{\mathop{\mathbf{C}}}\,)\}\] \[=\overset{\to }{\mathop{\mathbf{A}}}\,\cdot \{\overset{\to }{\mathop{\mathbf{B}}}\,\times \overset{\to }{\mathop{\mathbf{A}}}\,+\overset{\to }{\mathop{\mathbf{0}}}\,+\overset{\to }{\mathop{\mathbf{B}}}\,\times \overset{\to }{\mathop{\mathbf{C}}}\,+\overset{\to }{\mathop{\mathbf{C}}}\,\times \overset{\to }{\mathop{\mathbf{A}}}\,+\overset{\to }{\mathop{\mathbf{C}}}\,\times \overset{\to }{\mathop{\mathbf{B}}}\,+\overset{\to }{\mathop{\mathbf{0}}}\,\}\] \[=[\overset{\to }{\mathop{\mathbf{A}}}\,\overset{\to }{\mathop{\mathbf{B}}}\,\overset{\to }{\mathop{\mathbf{A}}}\,]+[\overset{\to }{\mathop{\mathbf{A}}}\,\overset{\to }{\mathop{\mathbf{B}}}\,\overset{\to }{\mathop{\mathbf{C}}}\,]+[\overset{\to }{\mathop{\mathbf{A}}}\,\overset{\to }{\mathop{\mathbf{C}}}\,\overset{\to }{\mathop{\mathbf{A}}}\,]-[\overset{\to }{\mathop{\mathbf{A}}}\,\overset{\to }{\mathop{\mathbf{B}}}\,\overset{\to }{\mathop{\mathbf{C}}}\,]\] \[=0\]
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