Just divide [tex]\mathbf u[/tex] by its magnitude:
[tex]\|\mathbf u\|=\sqrt{5^2+6^2+(-1)^2}=\sqrt{62}[/tex]
So a unit vector in the direction of [tex]\mathbf u[/tex] is
[tex]\dfrac{\mathbf u}{\|\mathbf u\|}=\dfrac5{\sqrt{62}}\,\mathbf i+\dfrac6{\sqrt{62}}\,\mathbf j-\dfrac1{\sqrt{62}}\,\mathbf k[/tex]
Then multiply this by -1 to get a unit vector pointing in the opposite direction.
