Find unit vectors which makes an angle 60 degrees with both vectors {1, -1, 0} and {1, 0, -1}.

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For instance:

Reduce[{Simplify[VectorAngle[{1, -1, 0}, {u, v, w}] ==
                 VectorAngle[{1, 0, -1}, {u, v, w}] == 60 °,
                    {u, v, w} ∈ Reals], Total[{u, v, w}^2] == 1},
                       {u, v, w}] /. Equal[a_, b_] :> Equal[a, FullSimplify[b]]

$$\left(u=0\land v=-\frac{1}{\sqrt{2}}\land w=-\frac{1}{\sqrt{2}}\right)\lor \left(u=\frac{2 \sqrt{2}}{3}\land v=\frac{1}{3 \sqrt{2}}\land w=\frac{1}{3 \sqrt{2}}\right)$$


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