We can use the Gram-Schmidt process for this.
Let $v_{1} = a_{1} = (1,0,1)$ and $v_{2} = a_{2} - \frac{a_{2} \cdot v_{1}}{v_{1} \cdot v_{1}}v_{1} = (1,1,-1) - \frac{(1,1,-1) \cdot (1,0,1)}{(1,0,1) \cdot (1,0,1)}(1,0,1) = (1,1,-1) - \frac{0}{2}(1,0,1) =
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