Find all orthogonal $3\times 3$ matrices of the form... Find all orthogonal $3\times 3$ matrices of the form
\begin{bmatrix}a&b&0\\c&d&1\\e&f&0\end{bmatrix}

Using the fact that $A^TA$ = $I_n$, I set that all up and ended up with the following system of equations:
$$\left\{\begin{array}{l}a^2 + e^2 = 1\\
ab + ef = 0\\
b^2 + f^2 = 1\end{array}\right.$$
I know I can let things equal the sine and cosine of theta, but I'm not exactly sure how to write this answer down on paper. There has to be tons of possibilities, right? How many exactly?

We have $c=d=0$ and from $$a^2+b^2=1$$ $$e^2+f^2=1$$ $$2ab+2ef=0$$ you get by adding all three equations
$$(a+b)^2+(e+f)^2=2.$$
