A) \[\overrightarrow{C}\,\bot \,\overrightarrow{A}\]
B) \[\overrightarrow{C}\,\bot \,\overrightarrow{B}\]
C) \[\overrightarrow{C}\,\bot \,(\overrightarrow{A}+\overrightarrow{B})\]
D) \[\overrightarrow{C}\,\bot \,(\overrightarrow{A}\times \overrightarrow{B})\]
Correct Answer: D
Solution :
From the property of vector product, we notice that \[\overrightarrow{C}\] must be perpendicular to the plane formed by vector \[\overrightarrow{A}\] and \[\overrightarrow{B}\]. Thus \[\overrightarrow{C}\] is perpendicular to both \[\overrightarrow{A}\] and \[\overrightarrow{B}\] and \[(\overrightarrow{A}+\overrightarrow{B})\]vector also, must lie in the plane formed by vector \[\overrightarrow{A}\] and \[\overrightarrow{B}\]. Thus \[\overrightarrow{C}\] must be perpendicular to \[(\overrightarrow{A}+\overrightarrow{B})\] also but the cross product \[(\overrightarrow{A}\times \overrightarrow{B})\] gives a vector \[\overrightarrow{C}\] which can not be perpendicular to itself. Thus the last statement is wrong.
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