A) \[\text{B}{{\text{A}}^{\text{2}}}\text{cos}\,\]
B) \[\text{B}{{\text{A}}^{\text{2}}}\sin \,\]
C) \[\text{B}{{\text{A}}^{\text{2}}}\sin \,\,\text{cos}\,\]
D) Zero
Correct Answer: D
Solution :
\[(\overrightarrow{B}\times \overrightarrow{A})\cdot \overrightarrow{A}\] \[=B\,A\,\,\cos \theta \,\hat{n}\cdot \overrightarrow{A}\] \[=0\] Here \[\hat{n}\] is perpendicular to both \[\overrightarrow{A}\] and \[\overrightarrow{B}.\] Alternative: \[(\overrightarrow{B}\times \overrightarrow{A})\cdot \overrightarrow{A}\] Interchange the cross and dot, we have, \[(\overrightarrow{B}\times \overrightarrow{A})\cdot \overrightarrow{A}=\overrightarrow{\text{B}}\cdot (\overrightarrow{A}\times \overrightarrow{A})=0\] \[(\because \overrightarrow{A}\times \overrightarrow{A}=0)\] NOTE: The volume of a parallelepiped bounded by vectors \[\overrightarrow{A},\]\[\overrightarrow{B}\] and \[\overrightarrow{C}\] can be obtained by giving formula \[(\overrightarrow{A}\times \overrightarrow{B})\cdot \overrightarrow{C}.\]
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