Answer
$\frac{(2,-2,0)}{\sqrt{8}}$
Work Step by Step
We know that $ai+bj+ck=(a,b,c).$
We know that for a matrix
$
\left[\begin{array}{rrr}
a & b & c \\
d &e & f \\
g &h & i \\
\end{array} \right]
$
the determinant, $D=a(ei-fh)-b(di-fg)+c(dh-eg).$
A vector orthogonal to $u$ and $v$ is $u\times v$.
$u×v$ is the determinant of the matrix $\begin{bmatrix}
i& j & k \\
1& 1&-1\\
1&1 &1 \\
\end{bmatrix}
$
Thus $u×v=(2,-2,0).$
And a unit vector parallel to this is: $\frac{u\times v}{|u\times v|}=\frac{(2,-2,0)}{\sqrt{2^2+(-2)^2+0^2}}=\frac{(2,-2,0)}{\sqrt{8}}$
