# Thread: Find an orthogonal matrix

1. ## Find an orthogonal matrix

Find an orthogonal matrix whose first row is $( \frac {1}{3} , \frac {2}{3} , \frac {2}{3} )$

Solution so far:

I know that the rows of an orthogonal make up of an orthogonal basis.

But I can't remember what method I need to use to find it.

Find an orthogonal matrix whose first row is $( \frac {1}{3} , \frac {2}{3} , \frac {2}{3} )$

Solution so far:

I know that the rows of an orthogonal make up of an orthogonal basis.

But I can't remember what method I need to use to find it.
You need to solve:

$a + 2b + 2c = 0$ .... (1)

$a^2 + b^2 + c^2 = 1$ .... (2)

Since there are an infinite number of solutions and you only need a matrix, let c = 0.

Then

$a + 2b = 0$ .... (1')

$a^2 + b^2 = 1$ .... (2')

A solution to (1') and (2') is $a = -\frac{2}{\sqrt{5}}$ and $b = \frac{1}{\sqrt{5}}$.

So the elements of the second row in the matrix are $-\frac{2}{\sqrt{5}}$, $\frac{1}{\sqrt{5}}$, 0.

To get the elements of the third row, you could construct a unit vector normal to (1, 2, 2) and (-2, 1, 0) and use its components .....

For checking purposes, I get $-\frac{2}{3\sqrt{5}}$, $-\frac{4}{3\sqrt{5}}$, $-\frac{5}{3\sqrt{5}}$.