ROTATIONS IN SPACE OF EVEN DIMENSIONS. 175 



The product is a linear function of the two vectors perpendicular to 

 both. In particular the product is zero only when 



A-C = B-D = 0. 



that is, w^hen J/i and M2 are completely perpendicular. Thus the, 

 star product of two plane vectors vanishes when and only when they lie, 

 in the same plane or are completely perpendicular. 

 An interesting case occurs when 



AC = ± BD 9^0. 



For simplicity let A, B, C, D be unit vectors. Any unit vector in Mi 

 can be written 



A" = A cos a -\- B sina 



and similarly a vector in M2 can be written 



F = C cos /3 + Z) sin /3. 



The angle 6 between these vectors satisfies the relation 



cos d = A-C cos a cos /3 + 5 • Z) sin a sin j3 

 = A-C (cos a cos /3 ± sin a sin j8) 

 = .4-Ccos {a ± iS). 



The minimum value of 6 is given by 



cos = ± A ■ C. 



This is the angle between X and M2. Hence all vectors in Mi make 

 the same angle with M2 and vice versa. Since A X C and B X D are 

 completely perpendicular unit planes, we could choose the vectors 

 A, B, C, D such that .4 X C is the complement of £ X D in their 

 four dimensional space. If then A-C = BD, Mi * M2 is equal to its 

 complement, that is, is self-complementary. If AC ^ — BD, 

 Ml * Ml is the negative of its complement that is, it is anti-self- 

 complementary. For all vectors in either plane to make the same 

 angle with the other plane it is then necessary and sufficient that 

 Ml * 3/2 be self-complementary or anti-self -complementary. 



