## History of vector calculation

The vector calculus goes back to H. G. Grassmann and parallel to Hamilton. Grassmann published in 1844 the "Lineale Ausdehnungslehre". As a precursor Descartes and Möbius apply. The Irish mathematician William Rowan Hamilton (1805 - 1865) developed the theory of quaternions, which are considered as precursors of the vectors. The term scalar goes back to Hamilton. 1888 the Italian mathematician Giuseppe Peano (1858-1932) developed an axiomatic definition of a vector space.

## Scalar

Quantities that can be represented by a real number. Examples for scalar values are temperature, mass, ...

## Vector

Vectors, in addition to its value require the specification of a direction. In physics, e.g. Velocity, field strength, ... In general, a vector is not limited to three dimensions. In general it is an n-tuple of real numbers that is often listed as a column vector.

$\overrightarrow{v}=\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\\ \vdots \\ {v}_{n}\end{array}\right)$

## Linear Dependence

Two vectors are called linearly dependent or collinear if one can be converted by multiplication by a scalar in the other. The cross product of collinear vectors disappears.

## Orthogonality

Two vectors are called orthogonal if the dot product of two vectors vanishes. Geometrically, the vectors are perpendicular to each other then that is the angle enclosed by the vectors is 90°.

## Unit vector

Vectors of length 1 are called unit vectors. Each vector can be converted by normalizing into the unit vector by the vector is divided by its length.

## Multiple products

Multiple products of vectors are not associative in general. That the general rule is that the order in which the products are carried out is relevant.

In general is:

$\overrightarrow{u}\cdot \left(\overrightarrow{v}\cdot \overrightarrow{w}\right)\ne \left(\overrightarrow{u}\cdot \overrightarrow{v}\right)\cdot \overrightarrow{w}$

and

$\overrightarrow{u}\u2a2f\left(\overrightarrow{v}\u2a2f\overrightarrow{w}\right)\ne \left(\overrightarrow{u}\u2a2f\overrightarrow{v}\right)\u2a2f\overrightarrow{w}$

For the double vector product the development applies:

$\overrightarrow{u}\u2a2f\left(\overrightarrow{v}\u2a2f\overrightarrow{w}\right)=\overrightarrow{v}\cdot \left(\overrightarrow{u}\cdot \overrightarrow{w}\right)-\overrightarrow{w}\cdot \left(\overrightarrow{u}\cdot \overrightarrow{v}\right)$

The scalar triple product $\left(\overrightarrow{u}\u2a2f\overrightarrow{v}\right)\cdot \overrightarrow{w}$ is equal to the volume of the plane defined by the three vectors parallelepiped. The scalar triple product is positive if the three vectors form a right hand system.

Lagrangesche Identity:

$\left(\overrightarrow{m}\u2a2f\overrightarrow{u}\right)\cdot \left(\overrightarrow{v}\u2a2f\overrightarrow{w}\right)$$=\left(\overrightarrow{m}\cdot \overrightarrow{v}\right)\left(\overrightarrow{u}\cdot \overrightarrow{w}\right)-\left(\overrightarrow{u}\cdot \overrightarrow{v}\right)\left(\overrightarrow{m}\cdot \overrightarrow{w}\right)$

## Geometrical approches

Equation of the line through the two points P_{0} und P_{1} given by the vectors r_{0} und r_{1}:

$\overrightarrow{r}=\overrightarrow{{r}_{0}}+\lambda \left(\overrightarrow{{r}_{1}}-\overrightarrow{{r}_{0}}\right)$

Distance of two points P_{0} and P_{1}:

$\left|\overrightarrow{{r}_{1}}-\overrightarrow{{r}_{0}}\right|$