Chapter 11.3, Problem 3E

To determine

To fill: The blank in the statement “ $\Vert u\times v\Vert =$$\text{\_\_\_\_\_\_}$ ”.

Answer to Problem 3E

The complete statement is “ $\Vert u\times v\Vert =\Vert u\Vert \Vert v\Vert \text{sin}\theta$ .”

Explanation of Solution

Given information:

The statement “ $\Vert u\times v\Vert =$$\text{\_\_\_\_\_\_}$ ”.

Consider the provided statement “ $\Vert u\times v\Vert =$$\text{\_\_\_\_\_\_}$ ”.

Recall that the magnitude of the cross product of any two vectors in a space is the product of the magnitudes of the two vectors and the sine angle between them. If there are two vectors in a space say $a=a_{1}i+a_{2}j+a_{3}k$ and $b=b_{1}i+b_{2}j+b_{3}k$ which have their magnitudes as $\Vert a\Vert$ and $\Vert b\Vert$ respectively, then the magnitude of their cross product is,

  $\Vert a\times b\Vert =\Vert a\Vert \Vert b\Vert \text{sin}\theta$

Apply it,

So, the magnitude of the given two vectors $u$ and $v$ in the space will be evaluated as,

  $\Vert u\times v\Vert =\Vert u\Vert \Vert v\Vert \text{sin}\theta$

Thus, the complete statement is $\Vert u\times v\Vert =\Vert u\Vert \Vert v\Vert \text{sin}\theta$ .