mathematics-physics-wiki/docs/en/physics/mathematical-physics/vector-analysis/vectors.md

Vectors and geometry

Axiomatic geometry

The defining property of axiomatic geometry is that it can be introduced without any reference to a coordinate system. The 5 postulates of classical geometry are listed below.

1. A straight line segment can be drawn between any pair of two points.
2. A straight line segment can be extended indefinitely into a straight line.
3. A line segment is the radius of a circle with one of the end points as its center.
4. All right angles are congruent.

The fifth postulate as formulated below is only valid for Euclidean geometry; flat space informally.

5. Given in a plane, a line and a point not on that line there is only one line through that point that does not intersect with the other line.

Vectors

Referring to linear algebra section vector spaces for the axioms of the Euclidean vector space and its vector definitions. Some vector products in 3 dimensional Euclidean space are defined below

Definition: the Euclidean scalar product of \mathbf{u}, \mathbf{v} \in \mathbb{R}^3 is given by

\langle \mathbf{u}, \mathbf{v} \rangle := |\mathbf{u}| |\mathbf{v}| \cos \varphi,

with \|\mathbf{u}\| and \|\mathbf{v}\| the length of \mathbf{u} and \mathbf{v} and the \varphi the angle between \mathbf{u} and \mathbf{v}.

It follows than that for \mathbf{v} = \mathbf{u} we have

|\mathbf{u}| = \langle \mathbf{u}, \mathbf{u} \rangle.

Definition: the Euclidean cross product of \mathbf{u}, \mathbf{v} \in \mathbb{R}^3 is given by

|\mathbf{u} \times \mathbf{v}| := |\mathbf{u}| |\mathbf{v}| \sin \varphi,

with \|\mathbf{u}\| and \|\mathbf{v}\| the length of \mathbf{u} and \mathbf{v} and the \varphi the angle between \mathbf{u} and \mathbf{v}. Defining the area of a parallelogram span by \mathbf{u} and \mathbf{v}. The normal direction of the surface is obtained by not taking the length of the cross product.

The scalar and cross product can be combined obtaining a parallelepiped spanned by three 3-dimensional vectors.

Definition: the Euclidean scalar triple of \mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^3 is given by

\langle \mathbf{u}, \mathbf{v}, \mathbf{w} \rangle := \langle \mathbf{u}, \mathbf{v} \times \mathbf{w} \rangle,

defining the volume of a parallelepiped spanned by \mathbf{u}, \mathbf{v} and \mathbf{w}.

Let J be a 3 \times 3 matrix given by J = (\mathbf{u}^T, \mathbf{v}^T, \mathbf{w}^T), the Euclidean scalar product may also be defined as

\langle \mathbf{u}, \mathbf{v}, \mathbf{w} \rangle = \det (J),

with \mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^3. We also have that

\langle \mathbf{u}, \mathbf{v}, \mathbf{w} \rangle^2 = \det (J^TJ).