Wacker Art Vectors Wappen der Familie Wacker
Ship entering the harbor.
Bild: "Ship Entering Harbor"


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...Though several people might be credited with conceiving the idea of "directed number", Hermann Grassmann, in his book of 1844, developed the idea with precision and completeness that far surpassed the work of anyone else at the time...
David Hestenes

..Grassmann looked for rules for combining vectors which would fully describe the geometrical properties of directed line segments. He noticed that two directed line segments connected end to end determined a third, which may be regarded as their sum...
David Hestenes

Den ersten Anstoss gab mir die Betrachtung des Negativen in der Geometrie; ich gewöhnte mich, die Strecken AB und BA als entgegengesetzte Größen aufzufassen;
Hermann Grassmann - Die Ausdehnungslehre - Vorrede.

The first impulse was the consideration of the negative in geometry; I got the habit to view the distances AB and BA as opposite values;
Hermann Grassmann - Die Ausdehnungslehre - Vorrede

distance AB = - distance BA

...woraus dann hervorging, dass, wenn A,B,C Punkte einer geraden Line sind, dann auch allemal AB + BC = AC sei, sowohl wenn AB und BC gleichbezeichnet sind...
Hermann Grassmann - Die Ausdehnungslehre - Vorrede

...from which it then emerged that if A, B, C are points of a straight line, then also always AB + BC = AC, both if AB and BC are labeled the same...
Hermann Grassmann - Die Ausdehnungslehre - Vorrede

...als auch wenn entgegengesetzt bezeichnet, d.h. wenn C zwischen A und B liegt. In dem letzteren Falle waren nun AB und BC nicht als blosse Längen aufgefasst, sondern an ihnen zugleich ihre Richtung festgehalten, vermöge deren sie eben einander engegengesetzt waren. So drängt sich der Unterschied auf zwischen der Summe der Längen und zwischen der Summe solcher Strecken, in denen zugleich eine Richtung festgehalten war.
Hermann Grassmann - Die Ausdehnungslehre - Vorrede

...as well as when labeled opposite, that means C is between A and B. In the latter case, AB and BC were not taken to be pure lengths, but also considering their direction, which is reverse to one another. The difference between the sum of the lengths and the sum of those distances in which the direction was considered becomes obvious.
Hermann Grassmann - Die Ausdehnungslehre - Vorrede

Hieraus ergab sich die Forderung, den letzten Begriff der Summe nicht bloss für den Fall, dass die Strecken gleich- oder entgegengesetzt-gerichtet waren, sondern auch für jeden anderen Fall festzustellen. Dies kann aufs einfachste geschehen in dem das Gesetz AB + BC = AC sei, auch dann noch festgehalten wurde, wenn A,B,C nicht in einer geraden Linie waren.
Hermann Grassmann - Die Ausdehnungslehre - Vorrede

This resulted in the requirement to use the last concept of a sum, not only for the case when the routes were directed in the same direction or opposite, but also for every other case. This can be done in the simplest way when the law AB + BC = AC is also applied, when A, B, C are not in a straight line.
Hermann Grassmann - Die Ausdehnungslehre - Vorrede


This page is based on the concepts developed on my geometric algebra page and on my linear algebra page.

Column Vectors of a Matrix

The column vectors of a matrix can be used to span a vector space.

Multiplying the column vector x with a matrix A will result in a column vector b.

If all three column vectors are linear independent, they span the whole ℝ3.

If only two column vectors are linear independent, they span a plane in ℝ3..

If only one column vector is linear independent, the multiple of this vector form a line through ℝ3.

Frames and Reciprocal Frames

Any set of lineare independent vectors form a basis.

Frame Vectors

The defining vectors do not have to be orthonormal vectors.

The volume element for this vectors is defined as follows:

Volume Element

The reciprocal vector ei is defined by the following formula. The check ei is left out on the right hand side of the formular.

Reciprocal Vector

If i = j the Kronecker delta is 1 in all other cases it is 0. That means that ei is orthogonal to all other vectors ej with the exception of ei.

Kronecker Relation

The concept of reciprocal frames is also known as covariant and contravariant vectors.

Covariant and Contravariant Vectors

Contravariant vectors are vectors with upper indices vi.

Covariant vectors are vectors with lower indices vi.

Example two Vectors in the Plane

e1 and e2 are orthogonal unit vectors and build a base of ℝ2.

v1 = x1e1+ y1e2

v2 = x2e1+ y2e2


v1·v1 = 1;

v2·v2 = 1;

v1·v2 = 0;    v1 and v2 are orthogonal.

v2·v1 = 0;    v2 and v1 are orthogonal.

Fig.: Covariant and Contravariant Vectors.

v1 = v1;    v2 = v2;

v1 = w1;    v2 = w2;

Bild: "Finnlines"

Axiomatic Definition of a Vector Space

Axioms of a vector space.


A vector space 𝕍 is a set of objects called vectors.

There are two operations defined on 𝕍:
Scalar multiplication av and vector addition: v+w.

There is a special member 0 ∈ 𝕍.

The following Axioms are satisfied for all vectors u,v,w ∈ 𝕍 and all scalars a,b ∈ ℝ


av ∈ 𝕍, v+w ∈ 𝕍

𝕍 is closed under scalar multiplication and vector addition.


v+w = w+v

Vector addition is commutative.


(u+v)+w = u+(v+w)  

Vector addition is associative.


v+0 = v

0 is the additive identity


0v = 0


1v = v

1 is the multiplicative identity.


a(bv) = (ab)v


a(v+w) = av + aw

Scalar multiplication is distributiv over vector addition.


(a+b)v = av + bv

Travemünde Harbor
Picture: "Travemünde"

The next page is about rotation matrices.

18. Dezember 2017 Version 1.0
Copyright: Hermann Wacker Uhlandstraße 10 D-85386 Eching bei Freising Germany Haftungsausschluß