Question

Two vectors $\overrightarrow{A}$ & $\overrightarrow{B}$ are shown in figure. Which of the following statements is NOT true about $\overrightarrow{A}\times \overrightarrow{B}$?

• A. It is a vector that points into the page
• B. It has a magnitude that is less than or equal to $12$
• C. It has no component in the plane of the page
• D. The angle it makes with $\overrightarrow{B}$ is less than the angle it makes with $\overrightarrow{A}$
• E. It is the same as $-\overrightarrow{B}\times \overrightarrow{A}$

Answer 1

The correct option is D The angle it makes with $\overrightarrow{B}$ is less than the angle it makes with $\overrightarrow{A}$

The cross product of two vectors is given by:

$\overrightarrow{R}=\overrightarrow{A}\times \overrightarrow{B}=AB\sin \theta \hat{n}$

where $A=3$ magnitude of vector A

$B=4$ magnitude of vector B

$\theta =$ smaller angle between $\overrightarrow{A}$ and $\overrightarrow{B}$

$\hat{n}=$ unit vector in the direction of $\overrightarrow{A}\times \overrightarrow{B}$

(A) The direction of $\overrightarrow{R}$ is given by Right hand screw rule, if we use this rule we find that $\overrightarrow{R}$ points into the page.

(B) Now $\overrightarrow{R}=3\times 4\sin \theta =12\sin \theta$,

As maximum value of $\sin \theta$ is 1, therefore $R\le 12$.

(C) As $\overrightarrow{R}$ is not in the plane of page (perpendicular to the page) therefore it cannot has a component in the plane of the page.

(D) $\overrightarrow{R}$ is perpendicular to both $\overrightarrow{A}$ and $\overrightarrow{B}$, therefore given statement is false.

(E)$\overrightarrow{A}\times \overrightarrow{B}=AB\sin \theta \hat{n}$, ..............eq1

and $\overrightarrow{B}\times \overrightarrow{A}=BA\sin \theta \hat{m}$, .............eq2,

By eq1 and eq2:

$\hat{n}=-\hat{m}$ (as both are in opposite direction)

$\overrightarrow{A}\times \overrightarrow{B}=AB\sin \theta \widehat{(}-\hat{m})$

$\overrightarrow{A}\times \overrightarrow{B}=-BA\sin \theta \hat{m}$

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