Chapter 10.3, Problem 69E

To determine

To Prove: $\Vert u\times v\Vert =\Vert u\Vert \Vert v\Vert$ if $u$ and $v$ are orthogonal.

Explanation of Solution

Given information:

  $\Vert u\times v\Vert =\Vert u\Vert \Vert v\Vert$

Calculation:

Calculate cross product of $u$ and $v$ .

  $\Vert u\times v\Vert =\Vert u\Vert \Vert v\Vert \sin \theta$

In above equation if the value of angle is $90°$ then the above equation will be same as the equation to be proved.

Substitute $90°$ for $\theta$ in above equation.

  $\begin{array}{l}\Vert u\times v\Vert =\Vert u\Vert \Vert v\Vert \sin 90°\\ \Vert u\times v\Vert =\Vert u\Vert \Vert v\Vert 1\\ \Vert u\times v\Vert =\Vert u\Vert \Vert v\Vert \end{array}$

Hence, proved