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 $90^{\circ}$ .

Example

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

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 $| \vec{u} | = 1$ , $| \vec{v} | = 1$ .

Example

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

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

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