Question

If the angle between the vectors $\vec{ A }$ and $\vec{ B }$ is $\theta$, the value of the product $(\vec{ B } \times \vec{ A }) \cdot \vec{ A }$ is equal to

• A. $B A^{2} \cos \theta$
• B. $B A^{2} \sin \theta$
• C. $B A^{2} \sin \theta \cos \theta$
• D. Zero

Answer 1

Answer: (D)

$(\vec{ B } \times \vec{ A }) \cdot \vec{ A } $
$=B A \cos \theta \vec{ n } \cdot \vec{ A } $
$=0$
Here $\hat{ n }$ is perpendicular to both $\vec{ A }$ and $\vec{ B }$.
Alternative: $(\vec{ B } \times \vec{ A }) \cdot \vec{ A }$
Interchange the cross and dot, we have,
$(\vec{B} \times \vec{A}) \cdot \vec{A}=\vec{B} \cdot(\vec{A} \times \vec{A})=0 $
$(\because \vec{A} \times \vec{A}=0)$
NOTE: The volume of a vecbounded by vectors $\vec{ A }, \vec{ A }$ and $\vec{ A }$ can be obtained by giving formula $(\vec{ A } \times \vec{ A }) \cdot \vec{ A }$