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


..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


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


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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.

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

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