Euclidean spaces Key concepts

Dimension

The dimension is the number of independent coordinates needed to locate a Point. A Line is $1$D, a Plane is $2$D, and Space (3D) is $3$D.

Coordinates

In ${\mathbb{R}}^n$, a point is represented by an ordered list

$$(x_1,x_2,\dots ,x_n).$$

Coordinates are labels, not the geometry itself: changing coordinates should not change distances or angles.

Vectors

A vector represents displacement, direction, and magnitude. The vector from $a$ to $b$ is

$$\mathbf{v}=b-a.$$

This is the language of position, Velocity, and Vector Calculus index.

Metric and distance

The Euclidean metric gives the dot product

$$\mathbf{u}\cdot \mathbf{v}=\sum _{i=1}^n{u}_i{v}_i.$$

It defines length

$$\Vert \mathbf{v}\Vert =\sqrt{\mathbf{v}\cdot \mathbf{v}}$$

and distance

$$d(p,q)=\sqrt{\sum _{i=1}^n(p_i-q_i{)}^2}.$$

Basis

A basis is a set of independent directions from which every vector can be built. In ${\mathbb{R}}^3$, the standard basis is usually $\hat{\mathbf{i}},\hat{\mathbf{j}},\hat{\mathbf{k}}$.

Flatness

Euclidean spaces have zero intrinsic Curvature. They are the local model for Manifolds, even when the full space is globally curved.

Inner products and norms

Euclidean geometry is powered by the dot product:

$$\mathbf{u}\cdot \mathbf{v}=\sum _i{u}_i{v}_i.$$

It gives lengths and angles:

\|\mathbf{v}\|=\sqrt{\mathbf{v}\cdot\mathbf{v}}, \qquad \cos heta=rac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}.

This is why Euclidean space is the default playground for Vector Calculus index, mechanics vectors, and basic geometry.