Take the cross product, then normalize the result.
[tex]\mathbf a\times\mathbf b=\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\1&7&1\\1&11&1\end{vmatrix}=-4\,\mathbf i+4\,\mathbf k[/tex]
This has norm
[tex]\|\mathbf a\times\mathbf b\|=\sqrt{(-4)^2+4^2}=4\sqrt2[/tex]
and so a unit vector orthogonal to both given vectors is
[tex]\dfrac{\mathbf a\times\mathbf b}{\|\mathbf a\times\mathbf b\|}=-\dfrac1{\sqrt2}\,\mathbf i+\dfrac1{\sqrt2}\,\mathbf k[/tex]
An equally correct answer would be the negative of this vector, since [tex]\mathbf b\times\mathbf a=-\mathbf a\times\mathbf b[/tex].
