There are two possible orthogonal transformations of $\mathbb{R}^2$ that leave the origin fixed and send the point $(0,13)$ to $(5,12)$. Find their matrices and describe them geographically.
Can anyone explain to me how I'd go about working this out? Sorry but I know nothing about orthogonal transformations and nowhere seems to explain how I can work them out well enough...
Hint: Consider the action on a basis.
You want $ (0,1) $ to get mapped to $( \frac{5}{13}, \frac {12}{13})$.
Since you want an orthogonal transformation, $(1,0)$ gets mapped to $(-\frac{12}{13}, \frac{5}{13})$.
Which matrix does this>