## Finding the missing vector such that the matrix is orthonormal

1. The problem statement, all variables and given/known data
A matrix is orthonormal if the magnitude of its vectors = 1 and all vector pairs are perpendicular, that is, their dot product = 0. Find the missing vector which would make the following matrix orthonormal:
$$\begin{bmatrix} \frac{1}{\sqrt{5}} & x \\ \frac{2}{\sqrt{5}} & y \\ \end{bmatrix}$$
Most textbooks use $$x_1 and x_2$$ but I find x and y easier
2. Relevant equations
3. The attempt at a solution
$$\frac{1}{\sqrt{5}}x + \frac{2}{\sqrt{5}} = 0 ... space here ... \sqrt{x^2 + y^2} = 1$$
I tried setting x = y and I got
$$\frac{1}{\sqrt{5}}x = -\frac{2}{\sqrt{5}}$$
But that's as far as I could with that problem.
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 I solved this problem.
 I've solved this problem.

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