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)

$(\vec{B}\times \vec{A})\cdot \vec{A}$
$=BA\cos \theta \vec{n}\cdot \vec{A}$
$=0$
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 $(\vec{A}\times \vec{A})\cdot \vec{A}$