# Vector Calculus

## 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.

$\stackrel{\to }{v}=\left(\begin{array}{c}{\mathrm{v1}}_{}\\ {v}_{2}\\ ⋮\\ {v}_{n}\end{array}\right)$

## 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.

## Calculation rules for vectors

### Multiplication of a vector with a scalar

The multiplication of a vector by a scalar positive λ only changes the length of the vector and not direction. In the multiplication with a scalar, the negative direction of the vector changes in the opposite direction.

${\lambda }\cdot \stackrel{\to }{v}=\left(\begin{array}{c}{\lambda }\cdot {v}_{1}\\ {\lambda }\cdot {v}_{2}\\ ⋮\\ {\lambda }\cdot {v}_{n}\end{array}\right)$

For the multiplication of a vector by a scalar λ the distributive law holds.

${\lambda }\cdot \left(\stackrel{\to }{v}+\stackrel{\to }{w}\right)={\lambda }\cdot \stackrel{\to }{v}+{\lambda }\cdot \stackrel{\to }{w}$

### Addition of vectors

The addition of vectors is done in Cartesian coordinates componentwise. The vector addition is commutative and associative.

$\stackrel{\to }{v}+\stackrel{\to }{w}=\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\\ ⋮\\ {v}_{n}\end{array}\right)+\left(\begin{array}{c}{w}_{1}\\ {w}_{2}\\ ⋮\\ {w}_{n}\end{array}\right)=\left(\begin{array}{c}{v}_{1}+{w}_{1}\\ {v}_{2}+{w}_{2}\\ ⋮\\ {v}_{n}+{w}_{n}\end{array}\right)$

Geometrically, the resulting vector can be constructed by one of the vectors is parallel shifted to the other end point of the vector. The connection from the start point of the first vector to the end point of the second vector is the resultant of vector of vector addition.Graphical vector addition  ### Subtraction of vectors

Subtraction of vectors is carried out as by the addition of the components are subtracted.

$\stackrel{\to }{v}-\stackrel{\to }{w}=\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\\ ⋮\\ {v}_{n}\end{array}\right)-\left(\begin{array}{c}{w}_{1}\\ {w}_{2}\\ ⋮\\ {w}_{n}\end{array}\right)=\left(\begin{array}{c}{v}_{1}-{w}_{1}\\ {v}_{2}-{w}_{2}\\ ⋮\\ {v}_{n}-{w}_{n}\end{array}\right)$

Geometrically, the construction similar to the addition of only the vector will be mirrored to the negative sign in the direction.Graphical vector subtraction  ### Dot Product (inner Product) of vectors

The dot product is defined as the product of the components and the sum of these products. The scalar product is not of the order-dependent (commutative). The scalar product name comes from the fact that the result is a scalar and not a vector.

$\stackrel{\to }{v}\cdot \stackrel{\to }{w}=\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\\ ⋮\\ {v}_{n}\end{array}\right)\cdot \left(\begin{array}{c}{w}_{1}\\ {w}_{2}\\ ⋮\\ {w}_{n}\end{array}\right)$ $={v}_{1}\cdot {w}_{1}+{v}_{2}\cdot {w}_{2}+\dots +{v}_{n}\cdot {w}_{n}$$=\sum _{i=1}^{n}\left({v}_{i}\cdot {w}_{i}\right)$

The scalar product can be interpreted as the product of the geometric projection of a vector in the direction of the other vector. Physically it means that the product is formed only with the Kompnente of the vector that is effective in the direction of another vector.
The scalar product can be expressed geometrically. Here, φ the included angle of the vectors.Graphical inner product  $\stackrel{\to }{v}\cdot \stackrel{\to }{w}=|\stackrel{\to }{v}||\stackrel{\to }{w}|\mathrm{cos}\phi$

For the scalar product of the distributive law apply

$\stackrel{\to }{u}\cdot \left(\stackrel{\to }{v}+\stackrel{\to }{w}\right)=\stackrel{\to }{u}\cdot \stackrel{\to }{v}+\stackrel{\to }{u}\cdot \stackrel{\to }{w}$

and the commutative law

$\stackrel{\to }{v}\cdot \stackrel{\to }{w}=\stackrel{\to }{w}\cdot \stackrel{\to }{v}$

Dot Product calculator for vectors with 3 components.

$\stackrel{\to }{v}\cdot \stackrel{\to }{w}=\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\\ {v}_{3}\end{array}\right)\cdot \left(\begin{array}{c}{w}_{1}\\ {w}_{2}\\ {w}_{3}\end{array}\right)$

Input fields for the vector elements.

v1= w1=

v2= w2=

v3= w3=

### Length of a vector

The magnitude of a vector geometrically corresponds to the length of the vector.

$\left|\stackrel{\to }{v}\right|=$ $|$ $\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\\ ⋮\\ {v}_{n}\end{array}\right)$ $|$$=\sqrt{{v}_{1}^{2}+{v}_{2}^{2}+\dots +{v}_{n}^{2}}$$=\sqrt{\sum _{i=1}^{n}{v}_{i}^{2}}$

Calculator for the length of a vector with 3 components.

$|$ $\stackrel{\to }{v}$ $|$ $=$ $|$ $\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\\ {v}_{3}\end{array}\right)$ $|$

Input fields for the vector elements.

v1=

v2=

v3=

### Cross Product (outer product) of vectors

The vector product is defined in three-dimensional Euclidean vector space as follows.

$\stackrel{\to }{v}\cdot \stackrel{\to }{w}=\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\\ {v}_{3}\end{array}\right)⨯\left(\begin{array}{c}{w}_{1}\\ {w}_{2}\\ {w}_{3}\end{array}\right)$$=\left(\begin{array}{c}{v}_{2}{w}_{3}-{v}_{3}{w}_{2}\\ {v}_{1}{w}_{3}-{v}_{3}{w}_{1}\\ {v}_{1}{w}_{2}-{v}_{2}{w}_{1}\end{array}\right)$

The vector product provides as a result a vector which is perpendicular to the plane spanned by the two vectors and the length of which corresponds to the surface area of ​​the clamped parallelogram.
The cross product can be expressed geometrically. Here, φ the included angle of the vectors and n is the vector perpendicular to the surface.

$v→⨯w→= |v→||w→|sinφn→$

The vector product is anti-commutative
$v→⨯w→ = -w→⨯v→$ Cross Product calculator for vectors with 3 components.

$\stackrel{\to }{v}\cdot \stackrel{\to }{w}=\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\\ {v}_{3}\end{array}\right)⨯\left(\begin{array}{c}{w}_{1}\\ {w}_{2}\\ {w}_{3}\end{array}\right)$

Input fields for the vector elements.

v1= w1=

v2= w2=

v3= w3=

### Multiplication of a vector with a matrix

The product of a matrix by a vector is a linear mapping. Explains the multiplication when the number of columns of the matrix is ​​equal to the number of elements of the vector. The result is a vector whose number of components equal to the number of rows of the matrix. That the example a matrix with two rows a vector always maps to a vector with two components.Graphical Matrix-Vector produkt $A\cdot \stackrel{\to }{v}=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1m}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2m}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nm}\end{array}\right)\cdot \left(\begin{array}{c}{v}_{1}\\ {v}_{2}\\ ⋮\\ {v}_{m}\end{array}\right)=$

$\left(\begin{array}{c}{a}_{11}{v}_{1}+{a}_{12}{v}_{2}+\dots +{a}_{1m}{v}_{m}\\ {a}_{21}{v}_{1}+{a}_{22}{v}_{2}+\dots +{a}_{2m}{v}_{m}\\ ⋮\\ {a}_{n1}{v}_{1}+{a}_{n2}{v}_{2}+\dots +{a}_{nm}{v}_{m}\end{array}\right)$

## 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:

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

and

$\stackrel{\to }{u}⨯\left(\stackrel{\to }{v}⨯\stackrel{\to }{w}\right)\ne \left(\stackrel{\to }{u}⨯\stackrel{\to }{v}\right)⨯\stackrel{\to }{w}$

For the double vector product the development applies:

$\stackrel{\to }{u}⨯\left(\stackrel{\to }{v}⨯\stackrel{\to }{w}\right)=\stackrel{\to }{v}\cdot \left(\stackrel{\to }{u}\cdot \stackrel{\to }{w}\right)-\stackrel{\to }{w}\cdot \left(\stackrel{\to }{u}\cdot \stackrel{\to }{v}\right)$

The scalar triple product $\left(\stackrel{\to }{u}⨯\stackrel{\to }{v}\right)\cdot \stackrel{\to }{w}$ 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(\stackrel{\to }{m}⨯\stackrel{\to }{u}\right)\cdot \left(\stackrel{\to }{v}⨯\stackrel{\to }{w}\right)$$=\left(\stackrel{\to }{m}\cdot \stackrel{\to }{v}\right)\left(\stackrel{\to }{u}\cdot \stackrel{\to }{w}\right)-\left(\stackrel{\to }{u}\cdot \stackrel{\to }{v}\right)\left(\stackrel{\to }{m}\cdot \stackrel{\to }{w}\right)$

## Geometrical approches

Equation of the line through the two points P0 und P1 given by the vectors r0 und r1:

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

Distance of two points P0 and P1:

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