MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 9 



In this direction cosine table we can "look up" the components of any unit 

 vector in terms of the other system. 



The matrix an is merely the matrix a,; with rows and columns inter- 

 changed, dji is called the conjugate of c,;. We shall denote the conjugate 

 of any matrix M by Mc. 



Obviously V is the vector sum of the 3 components (on the new system) 

 of each of its 3 components on the old system. 



/aiiVi -\- aT2 V2 + aTs FsX 

 (F)new ={a-'V -\- 



Fig. 33 — The direction cosines of Xj on X'l X'2 X'3. 



If the expression giving the components of V on the new system is de- 

 noted by V we may write 



V = aV 



conversely V — oT V 



Since xi is of unit length, the sum of the squares of its three components 

 (on the primed system) is unity. 



That is 



similarly 



and 



On + 012 + ai3 = 1 



O2I + O22 +^23= 1 • 



(4.3) 



031 + 032 + O33 



1 



Now Oc can be considered as a rotation similar to the rotation a. Con- 

 sequently their product aac is a similar rotation. Let us consider this 

 product. 



