Chapter 10, Problem 1P

(a)

To determine

Whether the given statement is true or false.

Answer to Problem 1P

This statement is true.

Explanation of Solution

Given:

Vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ are in opposite direction.

Formula Used:

Vector product is:

  $P=\overrightarrow{A}\times \overrightarrow{B}=AB\text{sin}\theta$

Calculation:

As vectors are in opposite direction, $\theta =180°$

  $\begin{array}{l}\sin \theta =\sin \left({180}^{\circ }\right)\\ \sin \left({180}^{\circ }\right)=0\end{array}$

Conclusion:

Thus, the given statement is true.

(b)

To determine

Whether the given statement is true or false.

Answer to Problem 1P

This statement is false.

Explanation of Solution

Given information :

Two vectors, $\overrightarrow{A}$ and $\overrightarrow{B}$ are perpendicular.

Formula Used:

Vector product is:

  $P=\overrightarrow{A}\times \overrightarrow{B}=AB\text{sin}\theta$

Calculation:

  $\begin{array}{l}\sin \theta =\sin \left({90}^{\circ }\right)\\ \sin \left({90}^{\circ }\right)=1\end{array}$

The cross product of $\overrightarrow{A}$ and $\overrightarrow{B}$ is a maximum.

Conclusion:

Thus, the given statement is false.

(c)

To determine

Whether, the given statement is true or false.

Answer to Problem 1P

This statement is true.

Explanation of Solution

Given information:

Magnitude of vector product of two non-zero vectors is known. Also, the magnitude of each vector is known.

Formula Used:

The angle between two vectors can be found as follows:

  $\theta ={\sin }^{-1}\left(\frac{\left|\overrightarrow{A}\times \overrightarrow{B}\right|}{AB}\right)$

Calculation:

From the given information, the angle can be found. Hence, the given statement is true.