We start this section with a definition.
Definition (D 1.4.1) A set of 3 linearly independent vectors r, r, and r in space is called a basis of the vector space. Any vector r of the vector space can be written in the form . The vectors r, r, and r are called basis vectors; the vector r is called a linear combination of r, r, and r. The real numbers , , and are called the coefficients of r with respect to the basis r, r, r. In crystallography the 2 basis vectors for the plane are mostly called a and b or and , and the 3 basis vectors of the space are a, b, and c or , , and .
The vector
connects the points
and , see Fig. 1.3.1. In Section 1.1 the coordinates ,
, and of a point have been introduced, see Fig. 1.1.1. We now
replace the section on the coordinate axis
by the vector
, on by
, and on by
. If and are given by
their columns of coordinates with respect to these coordinate axes, then
the vector
is determined by the column of
the three coordinate differences between the points and . These
differences are the vector coefficients of r:
As the point coordinates, the vector coefficients are written in a column. It is not always obvious whether a column of 3 numbers represents a point by its coordinates or a vector by its coefficients. One often calls this column itself a `vector'. However, this terminology should be avoided. In crystallography both, points and vectors are considered. Therefore, a careful distinction between both items is necessary.
An essential difference between the behaviour of vectors and points is provided by the changes in their coefficients and coordinates if another origin in point space is chosen:
Let be the new, the old origin, and o the column of coordinates of with respect to the old coordinate system: .
Then and the coordinates of and in the old coordinate system, are replaced by the columns and of the coordinates in the new coordinate system, see Fig.1.4.1.
From follows and , etc. Therefore, the coordinates of the points change if one chooses a new origin.
However, the coefficients of the vector do not change because of , etc.
Fig. 1.4.1 The coordinates of the points
and with respect to the old origin are x and y, with
respect to the new origin are
and
. From the
diagram one reads the equations
and

The rules 1., 2., and 5. of Section 1.3 (the others are then obvious) are expressed by:
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