Difference between revisions of "Vector space"
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The [[point space]] is a dual of the vector space because to each vector in vector space a pair of points in [[point space]] can be associated. | The [[point space]] is a dual of the vector space because to each vector in vector space a pair of points in [[point space]] can be associated. | ||
| − | Face normals, translation vectors, [[Patterson]] | + | Face normals, translation vectors, [[Patterson vector]]s and [[reciprocal lattice]] vectors are elements of vector space. |
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==See also== | ==See also== | ||
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* Chapter 8.1 in the ''International Tables for Crystallography Volume A'' | * Chapter 8.1 in the ''International Tables for Crystallography Volume A'' | ||
* [http://www.iucr.org/iucr-top/comm/cteach/pamphlets/22/index.html Matrices, Mappings and Crystallographic Symmetry], teaching pamphlet No. 22 of the [[International Union of Crystallography]] | * [http://www.iucr.org/iucr-top/comm/cteach/pamphlets/22/index.html Matrices, Mappings and Crystallographic Symmetry], teaching pamphlet No. 22 of the [[International Union of Crystallography]] | ||
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 13:18, 9 April 2008
For each pair of points X and Y in point space one can draw a vector r from X to Y. The set of all vectors forms a vector space. The vector space has no origin but instead there is the zero vector which is obtained by connecting any point X with itself. The vector r has a length which is designed by |r| = r, where r is a non–negative real number. This number is also called the absolute value of the vector. The maximal number of linearly independent vectors in a vector space is called the dimension of the space.
An essential difference between the behaviour of vectors and points is provided by the changes in their coefficients and coordinates if a different origin in point space is chosen. The coordinates of the points change when moving from an origin to the other one. However, the coefficients of the vector r do not change.
The point space is a dual of the vector space because to each vector in vector space a pair of points in point space can be associated.
Face normals, translation vectors, Patterson vectors and reciprocal lattice vectors are elements of vector space.
See also
- Chapter 8.1 in the International Tables for Crystallography Volume A
- Matrices, Mappings and Crystallographic Symmetry, teaching pamphlet No. 22 of the International Union of Crystallography