Question

If u and v are nonzero vectors in ${\mathrm{ℝ}}^3$, what is the geometric relationship between $u,v$ and $u\times v$?

Short answer

The relation between $u,v\mathrm{and}u\times v$is $||u\times v||=||u||||v||\sin \theta$,where θ is the angle between u and v.

Step-by-step solution

Step 1. Given Information 

If u and v are nonzero vectors in ${\mathrm{ℝ}}^3$, what is the geometric relationship between $u,v$ and $u\times v$?

Step 2. u and v are nonzero vectors in ℝ3 with the same initial point.

By Lagrange’s identity

${||u\times v||}^2={||u||}^2{||v||}^2-{(u·v)}^2$

But since $u·v=||u||||v||\cos \theta$, we have

$$\begin{gathered} {||u\times v||}^2={||u||}^2{||v||}^2-{\left(||u||||v||\cos \theta \right)}^2 \\ {||u\times v||}^2={||u||}^2{||v||}^2(1-{\cos }^2\theta ) \\ {||u\times v||}^2={||u||}^2{||v||}^2{\sin }^2\theta \end{gathered}$$

Step 3. Then the relation between u,v and u×v is 

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

where θ is the angle between u and v.