Find vectors vertical to given vectors with certain length

Question

Given the vectors $\mathbf{u,v}$ in R³, determine all vectors that are vertical to $\mathbf{u}$ and $\mathbf{v}$ with length = 1

Every vector $\mathbf{x'}$ that is to be found must meet these two conditions:

• $\mathbf{x'} \cdot \mathbf{u} = \mathbf{x'} \cdot \mathbf{v}=0$
• $\mathbf{x'} \times \mathbf{u}= \mathbf{x'} \times \mathbf{v}=1$

My approach is to insert the given vector components into the first condition and get something like this
$ ax_1' + bx_2' + cx_3' =0 $

since the length of the vectors should be 1, there's

$||x'||=\sqrt{( x' \cdot x')}=\sqrt{x_1'^2+x_2'^2+x_3^3}$

is that the right way to go? I don't really know how to tackle this

Answer 1

If you want it to be perpendicular to both $\mathbf{u}$ and $\mathbf{v}$ then take $ \mathbf{u} \times \mathbf{v}$. And then to normalize it (so that it has length 1) just divide this vector by its norm to get

$$\mathbf{x'}=\frac{\mathbf{u} \times \mathbf{v}}{|| \mathbf{u} \times \mathbf{v}||}$$