Finding orthogonal operators in $\mathbb{R}^2$

Question

Let $V = \mathbb{R}^2$ where $V$ is an inner product space with dot product. Let $v \in V$ be a unit vector.

I want to show there are exactly two orthogonal operators $T: V \to V$ such that $T((1,0)) = v$.

I have no idea where to start and any suggestions would be much appreciated!

Answer 1

Hint: What are the possibilities for $T(0,1)$? Once we know what $T$ does to $(1,0)$ and $(0,1)$, we know everything about $T$.

Answer 2

Let $e_1 = (1,0)$, $e_2 = (0,1)$ and $T(e_1)=v, T(e_2)=w$. You have $$1=\langle e_2,e_2\rangle = \langle T(e_2),T(e_2)\rangle = \|w\| ^2 $$ and $$0 = \langle e_1,e_2\rangle = \langle T(e_1),T(e_2)\rangle = \langle v,w\rangle $$

Solve the equations for $w$.