Chapter 9.7, Problem 5AYU

To determine

True or False, $\Vert u \times v\Vert = \Vert u\Vert \Vert v\Vert \cos \theta$, where $\theta$ is the angle between $\text{u}$ and $\text{v}$

Answer to Problem 5AYU

Solution:

False

Explanation of Solution

We know that, $\cos \theta = \frac{\text{u}.\text{v}}{\Vert \text{u}\Vert \Vert \text{v}\Vert }$

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

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

Hence, $\Vert \text{u} \times \text{v}\Vert \ne \Vert \text{u}\Vert \Vert \text{v}\Vert \cos \theta ,$

Therefore the answer is False.