15.2: Euclidean Space

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Let us repeat the construction of metric \(d_2\) in space.

Suppose that \(\mathbb{R}^3\) denotes the set of all triples \((x,y,z)\) of real numbers. Assume \(A=(x_A,y_A,z_A)\) and \(B=(x_B,y_B,z_B)\) are arbitrary points in \(\mathbb{R}^3\). Define the metric on \(\mathbb{R}^3\) the following way:

\(AB := \sqrt{(x_A-x_B)^2+(y_A-y_B)^2+(z_A-z_B)^2}.\)

The obtained metric space is called Euclidean space.

The subset of points in \(\mathbb{R}^3\) is called plane if it can be described by an equation

\(a\cdot x+b\cdot y+c\cdot z+d=0\)

for some constants \(a\), \(b\), \(c\), and \(d\) such that at least one of values \(a\), \(b\) or \(c\) is distinct from zero.

It is straightforward to show the following:

Any plane in the Euclidean space is isometric to the Euclidean plane.
Any three points in the space lie on a plane.
An intersection of two distinct planes (if it is nonempty) is a line in each of these planes.

These statements make it possible to generalize many notions and results from Euclidean plane geometry to the Euclidean space by applying plane geometry in the planes of the space.