Question

For the vectors $\overrightarrow{A}$and $\overrightarrow{B}$ making an angle $\theta$. Which one of the following relations is correct?

• A. $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{A}$
• B. $\overrightarrow{A}\times \overrightarrow{B}=AB\sin \left(\theta \right)$
• C. $\overrightarrow{A}.\overrightarrow{B}=AB\cos \left(\theta \right)$
• D. $\overrightarrow{A}\times \overrightarrow{B}=-\overrightarrow{B}\times \overrightarrow{A}$

Answer 1

The correct option is D

$\overrightarrow{A}\times \overrightarrow{B}=-\overrightarrow{B}\times \overrightarrow{A}$

The explanation for the correct option(s):

D: $\overrightarrow{\mathbf{A}}\mathbf{\times }\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{-}\overrightarrow{\mathbf{B}}\mathbf{\times }\overrightarrow{\mathbf{A}}$

One of the properties of cross-product is anti-commutative.

The explanation for the incorrect option(s):

A: $\overrightarrow{\mathbf{A}}\mathbf{\times }\overrightarrow{\mathbf{B}}\mathbf{=}\overrightarrow{\mathbf{B}}\mathbf{\times }\overrightarrow{\mathbf{A}}$

One of the properties of cross-product is commutative.

B: $\overrightarrow{\mathbf{A}}\mathbf{\times }\overrightarrow{\mathbf{B}}\mathbf{=}\mathit{A}\mathit{B}\mathit{s}\mathit{i}\mathit{n}\left(\theta \right)$

If $\theta$ be the angle between two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$, the cross product will be $\overrightarrow{A}\times \overrightarrow{B}=AB\sin \left(\theta \right)$

C: $\overrightarrow{\mathbf{A}}\mathbf{.}\overrightarrow{\mathbf{B}}\mathbf{=}\mathit{A}\mathit{B}\mathit{c}\mathit{o}\mathit{s}\left(\theta \right)$

If $\theta$ be the angle between two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$, the dot product will be $\overrightarrow{A}.\overrightarrow{B}=AB\cos \left(\theta \right)$

Hence, So it concludes that the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ making an angle $\theta$ if the cross product of $\overrightarrow{A}$ and $\overrightarrow{B}$ is anti-commutative.