Q:

Find a unit vector that is orthogonal to both $u = (1, 0, 1)$ and $v = (0, 1, 1)$.

Find a unit vector that is orthogonal to both $u = (1, 0, 1)$ and $v = (0, 1, 1)$.
I am lost on how to find one orthogonal to both.. I used the unit vector rule:
vector $= (1/||v||)v$ on both vectors to get $(-1/\sqrt2,0,-1/\sqrt{2})$ and $(0,-1/\sqrt2,-1/\sqrt2)$ after making them negative to make them oppositely directed.. but I can't figure out how to find one that is orthogonal to both.. 

A:

$u$ and $v$ are orthogonal if $u\cdot v=0$.
you want a vector $(a,b,c)$ such that $(a,b,c)\cdot (1,0,1)=0$ and $(a,b,c)\cdot (0,1,1)=0$.  That is $a+c=0$ and $b+c=0$.  There are many possible solutions for $a,b,c$ which satisfy both of these equations.  For example, $a=1,b=1,c=-1$ works.
Now multiply $(1,1,-1)$ by $1/$(its norm) to get the unit vector $(1/\sqrt 3,1/\sqrt 3,-1/\sqrt 3)$.

