Q. 73 - Calculus

Question

Let u and v be vectors in ${\mathrm{ℝ}}^3$ such that $u·v=0$. Prove that if $\theta$ is the angle between u and v, then $\tan \theta =\frac{||u\times v||}{u·v}$

Step-by-Step Solution

Answer

Hence, we prove that if $\theta$ is the angle between u and v, then $\tan \theta =\frac{||u\times v||}{u·v}$

Step 1. Given Information

Let u and v be vectors in ${\mathrm{ℝ}}^3$ such that $u·v=0$. Prove that if $\theta$ is the angle between u and v, then $\tan \theta =\frac{||u\times v||}{u·v}$

Step 2. As we know that if u, v, and u × v form a right-handed triple.

Then, $||u\times v||=||u||||v||\sin \theta \dots \dots \dots \dots \mathrm{Equation} 1$

u, v, and $u·v$ form a left-handed triple.

$u·v=||u||||v||\cos \theta \dots \dots \dots \dots \mathrm{Equation} 2$

Step 3. Divide the equation 1 and 2

$$\begin{gathered} \frac{||u\times v||}{u·v}=\frac{||u||||v||\sin \theta }{||u||||v||\cos \theta } \\ \frac{||u\times v||}{u·v}=\frac{\sin \theta }{\cos \theta } \\ \frac{||u\times v||}{u·v}=\tan \theta \\ \tan \theta =\frac{||u\times v||}{u·v} \end{gathered}$$