Geometry from Inner Products

Geometric notions such as lengths and angles can be generalised over vector spaces provided that they have an inner product (which can only happen if they are fixed over a real or complex field). I.e., such notions only exist in an inner product space.

Length §

Length ^formula1

The length (or norm) of a vector, $\vec{v}$, in an inner product space is defined as:

$$\Vert \vec{v}\Vert =\sqrt{⟨\vec{v},\vec{v}⟩}$$

Distance §

Distance

The distance between two vectors, $\vec{u}$ and $\vec{v}$ in an inner product space is defined as:

$$\text{dist}(\vec{u},\vec{v})=\Vert \vec{v}-\vec{u}\Vert$$

Angle §

Angle

The angle between two vectors $\vec{u},\vec{v}$ in a real inner product space is defined as:

$$\cos (\theta )=\frac{⟨\vec{u},\vec{v}⟩}{\Vert \vec{u}\Vert \Vert \vec{v}\Vert }\theta \in [0,\pi ]$$

In a complex product space:

$$\cos (\theta )=\frac{\text{Re}(⟨\vec{u},\vec{v}⟩)}{\Vert \vec{u}\Vert \Vert \vec{v}\Vert }\theta \in [0,\pi ]$$