Question

1. Find a unit vector that is orthogonal to both u and v.

Answer 1

Take the cross product of $\mathbf u$ and $\mathbf v$, then divide the cross product by its norm.

$\mathbf u\times\mathbf v=\begin{vmatrix}\langle1,0,0\rangle&\langle0,1,0\rangle&\langle0,0,1\rangle\\-8&-6&4\\17&-18&-1\end{vmatrix}$

$=\begin{vmatrix}-6&4\\-18&-1\end{vmatrix}\langle1,0,0\rangle-\begin{vmatrix}-8&4\\17&-1\end{vmatrix}\langle0,1,0\rangle+\begin{vmatrix}-8&-6\\17&-18\end{vmatrix}\langle1,0,1\rangle$

$=78\langle1,0,0\rangle+60\langle0,1,0\rangle+246\langle0,0,1\rangle$

$=\langle78,60,246\rangle$

which has norm

$\|\mathbf u\times\mathbf v\|=\sqrt{78^2+60^2+246^2}=30\sqrt{78}$

Then the unit vector is

$\dfrac{\mathbf u\times\mathbf v}{\|\mathbf u\times\mathbf v\|}=\left\langle\dfrac{\sqrt{78}}{30},\dfrac{\sqrt{78}}{39},\dfrac{41\sqrt{78}}{390}\right\rangle$

(and is unique up to its sign, meaning you can multiply this vector by -1 and still be correct)