Orthogonal unit vectors

[qwertyuiopasd]

If I know that {u,v,w} is an orthonormal basis, how do I find all the unit vectors that are orthogonal to u and u+v-w?

 

[Romsek]

[MATH]\text{A vector $z$ in the space spanned by $\{u,v,w\}$ is given by $z = a u + b v + c w,~a,b,c \in \mathbb{R}$}\\ \text{$u\cdot z = a u\cdot u + b u \cdot v + c u \cdot w = a \|u\|^2 + 0 + 0 = a$}\\ \text{Thus in order to be orthogonal to $u, ~a=0,~z =b v + c w,~b,c \in \mathbb{R}$}[/MATH]
[MATH]\text{Repeat this idea for $u+v-w$}[/MATH]

 

[LCKurtz]

Alternatively, notice that a vector normal to both u and u+v-w must be parallel to their cross product and go from there.