Chapter 11, Problem 11.1QQ

Which of the following statements about the relationship between the magnitude of the cross product of two vectors and the product of the magnitudes of the vectors is true? (a) $|\stackrel{\to }{A}\times \stackrel{\to }{B}|$ is larger than AB. (b) $|\stackrel{\to }{A}\times \stackrel{\to }{B}|$ is smaller than AB. (c) $|\stackrel{\to }{A}\times \stackrel{\to }{B}|$ could be larger or smaller than AB, depending on the angle between the vectors. (d) $|\stackrel{\to }{A}\times \stackrel{\to }{B}|$ could he equal to AB.

To determine

The statement that is true about the relationship between the magnitude of the cross product of two vectors and the product of the magnitude of the vector.

Answer to Problem 11.1QQ

Option(d).

Explanation of Solution

The magnitude of the cross product of two vector that is $\overrightarrow{\text{A}}$ and $\overrightarrow{\text{B}}$ is,

    $\left|\overrightarrow{\text{A}}\times \overrightarrow{\text{B}}\right|=AB\sin \theta$

Here, $\theta$ is the angle between the two vectors.

The value of $\left|\overrightarrow{\text{A}}\times \overrightarrow{\text{B}}\right|$ is a function of $\sin \theta$ so the value depends on the angle between the two vectors. As the angle varies between the two vectors, the magnitude of the cross product of the two vectors also changed. When, $\theta$ is $\frac{\pi }{2}$, $\sin \theta$ is $1$ and hence $\left|\overrightarrow{\text{A}}\times \overrightarrow{\text{B}}\right|$ equals $AB$.

Conclusion:

When both $\overrightarrow{A}$ and $\overrightarrow{B}$ are perpendicular,  $\left|\overrightarrow{\text{A}}\times \overrightarrow{\text{B}}\right|$ equals $AB$ Thus, option (d) is correct.

The value of $\sin \theta$ is never greater than $1$ and hence $\left|\overrightarrow{\text{A}}\times \overrightarrow{\text{B}}\right|$ is lesser than $AB$. Thus, option (a) is incorrect.

$\left|\overrightarrow{\text{A}}\times \overrightarrow{\text{B}}\right|$ is smaller than $AB$ for every other angle except $\frac{\pi }{2}$. Thus, option (b) is incorrect.

From option (b) and (c), $\left|\overrightarrow{\text{A}}\times \overrightarrow{\text{B}}\right|$ is couldn’t be always larger or smaller with regard to the angle. Thus, option (c) is incorrect.