At the end, you have (1, 0, 1), (0, 2, 0), and (-1, 0, 1) as orthogonal vectors and then divide the vectors by , 2, and to get unit vectors. That is correct even with this non-standard inner product. The last part asks you to write (2, 1, -1) as a linear combination of these basis vectors. That means you want to find numbers, a, b, and c, such that .

You could do this by solving the three equations , ,

But because this is an "orthonormal basis" there is a simpler way: take the dot product of [tex](\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}) with both sides of the equation. That will give

Take the dot product of (0, 1, 0) with both sides of the equation. That will give

Finally, take the dot product of with both side of the equation to get .