Space (3D)

Euclidean space

The Euclidean space of dimension three, denoted ${\mathbb{E}}^3$ or ${\mathbb{R}}^3$, is a geometric space in which three real numbers are required to determine the position of each point.

A point in 3D space is usually written

$$(x,y,z),$$

where the coordinates measure signed distance along three mutually perpendicular axes. The standard basis vectors are often written

$$\hat{\mathbf{i}}=(1,0,0),\hat{\mathbf{j}}=(0,1,0),\hat{\mathbf{k}}=(0,0,1).$$

Any vector can be decomposed as

$$\mathbf{v}=v_x\hat{\mathbf{i}}+v_y\hat{\mathbf{j}}+v_z\hat{\mathbf{k}}.$$

The distance between points $p=(x_1,y_1,z_1)$ and $q=(x_2,y_2,z_2)$ is

$$d(p,q)=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}.$$

This is the 3D version of Pythagoras.

Three-dimensional space is the natural setting for many physics quantities: position, Velocity, Momentum, force, electric fields, and magnetic fields. Vector Calculus index mostly begins here, using the Del operator ∇ to describe how scalar and vector fields vary throughout space.

Although ordinary space is well approximated by Euclidean geometry at human scales, general relativity models spacetime as a curved manifold, not a global Euclidean space.

Coordinates and vectors

A point in 3D Euclidean space is written

$$(x,y,z).$$

A displacement vector between two points is

$$\Delta \mathbf{r}=(x_2-x_1,y_2-y_1,z_2-z_1).$$

Its length is

$$|\Delta \mathbf{r}|=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}.$$

This is the default geometry assumed in first-year mechanics before relativity or curved coordinates enter.