Finding the missing vector such that the matrix is orthonormal

[g.lemaitre]

Homework Statement

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:
<br /> \begin{bmatrix}<br /> \frac{1}{\sqrt{5}} &amp; x \\<br /> \frac{2}{\sqrt{5}} &amp; y \\<br /> \end{bmatrix}
Most textbooks use x_1 and x_2 but I find x and y easier

Homework Equations

The Attempt at a Solution

<br /> \frac{1}{\sqrt{5}}x + \frac{2}{\sqrt{5}} = 0 ... space here ... <br /> \sqrt{x^2 + y^2} = 1<br />
I tried setting x = y and I got
<br /> \frac{1}{\sqrt{5}}x = -\frac{2}{\sqrt{5}}
But that's as far as I could with that problem.

 

[g.lemaitre]

I solved this problem.

 

[g.lemaitre]

I've solved this problem.