# Finding the missing vector such that the matrix is orthonormal

by g.lemaitre
Tags: matrix, missing, orthonormal, vector
 P: 274 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.
 P: 274 I solved this problem.
 P: 274 I've solved this problem.

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