International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 5-6

## Section 1.1.2.3. Orthonormal frames of coordinates – rotation matrix

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France

#### 1.1.2.3. Orthonormal frames of coordinates – rotation matrix

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An orthonormal coordinate frame is characterized by the fact that One deduces from this that the scalar product is written simply as

Let us consider a change of basis between two orthonormal systems of coordinates: Multiplying the two sides of this relation by , it follows that which can also be written, if one notes that variance is not apparent in an orthonormal frame of coordinates and that the position of indices is therefore not important, as

The matrix coefficients, , are the direction cosines of with respect to the basis vectors. Similarly, we have so that where T indicates transpose. It follows that so that The matrices A and B are unitary matrices or matrices of rotation and

 If the senses of the axes are not changed – proper rotation. If the senses of the axes are changed – improper rotation. (The right hand is transformed into a left hand.)

One can write for the coefficients giving six relations between the nine coefficients . There are thus three independent coefficients of the matrix A.