Orthogonal basis and orthonormal basis

We say that $B=\{\vec{u},\vec{v}\}$ is an orthogonal basis if the vectors that form it are perpendicular. In other words, $\vec{u}$ and $\vec{v}$ form an angle of $90^\circ$.

$\vec{u}=(3,0)$, $\vec{v}=(0,-2)$ form an orthogonal basis since the scalar product between them is zero and this a sufficient condition to be perpendicular: $$\vec{u}\cdot\vec{v}=3\cdot0+0\cdot(-2)=0$$

We say that $B=\{\vec{u},\vec{v}\}$ is an orthonormal basis if the vectors that form it are perpendicular and they have length $1$. Namely, $\vec{u}$ and $\vec{v}$ form an angle of $90^\circ$ and $|\vec{u}|=1$, $|\vec{v}|=1$.

$\vec{u}=(1,0)$, $\vec{v}=(0,-1)$ form an orthonormal basis since the vectors are perpendicular (its scalar product is zero) and both vectors have length $1$.

Perpendicular: $\vec{u}\cdot\vec{v}=1\cdot0+0\cdot(-1)=0$.

Unitary vectors (length 1): $|\vec{u}|=\sqrt{1^2+0^2}=\sqrt{1}=1$, $|\vec{v}|=\sqrt{0^2+(-1)^2}=\sqrt{1}=1$.