Vectors are objects which are encountered everywhere in crystallography: as distance vectors between atoms, as basis vectors of the coordinate system, as translation vectors of a crystal lattice, as vectors of the reciprocal lattice, etc. They are elements of the vector space which is studied by linear algebra and is an abstract space. However, vectors can be interpreted easily visually, see Fig.1.3.1:
|Fig. 1.3.1 Vector from point X to point Y. The vector represented by an arrow depends only on the relative but not on the absolute sites of the points. The 4 parallel arrows represent the same vector.|
For each pair of points X and Y one can draw the arrow from X to Y. The arrow is a representation of the vector r, as is any arrow of the direction and length of r, see Fig. 1.3.1. The set of all vectors forms the vector space. The vector space has no origin but instead there is the zero vector or o vector which is obtained by connecting any point with itself. The vector r has a length which is designed by , where r is a non-negative real number. This number is also called the absolute value of the vector. A formula for the calculation of can be found in Sections 1.6 and 2.6.
For such vectors some simple rules hold which can be visualized, e.g. by a drawing in the plane:
In particular, is a vector of length 1. Such a vector is called a unit vector. Further ; is the zero-vector with length 0. It is the only vector with no direction. is that vector which has the same length as r, , but opposite direction.
Fig. 1.3.3 Visualization of the associativity
of vector addition:
Definition (D 1.3.2) A set of vectors , ,
..., is called linearly independent if the equation
In the plane any 3 vectors r, r, and r
are linearly dependent because coefficients can always be found
such that not all zero and
Definition (D 1.3.2) The maximal number of linearly independent vectors in a vector space is called the dimension of the space.
As is well known, the dimension of the plane is 2, of the space is 3. Any 4
vectors in space are linearly dependent. Thus, if there are 3 linearly
independent vectors r, r, and r, then
any other vector r can be represented in the form
Such a representation is widely used, it will be considered in the next section.
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