Find all orthonormal bases of $\mathbb{R^2}$

Question

The question is short:

''Find all orthonormal bases of $\mathbb{R^2}.$

Recall. A basis $\left\{v_1,...,v_n\right\}$ of $V$ is said to be orthogonal if its elements are mutually perpendicular. If in addition each element of the basis has norm $1$, then the basis is called orthonormal.

How can we find these? Can you help; can you give a hint?

Answer 1

For $\theta \in \mathbb{R}$, any unit vector in $\mathbb{R}^2$ can be expressed as:

$$ u_1 = (\cos\theta, \sin\theta) $$

Can you find an orthonormal vector to $u_1$?

Answer 2

Hint

Define two vectors:$$v=[v_1\ \ v_2]\\u=[u_1\ \ u_2]$$and apply the orthogonality condition$$uv^T=v_1u_1+v_2u_2=0$$provided $v_1^2+v_2^2\ne 0$ and $u_1^2+u_2^2\ne 0$. What does it lead to? How many variables and how many equations to find them, remain there?