174 PHILLIPS AND MOORE. 



that is I -Ml and /• il/2 are commutative. Equation (25) is then satis- 

 fied and so 



il/i * il/2 = 0, 



since this equation was shown to be a consequence of (25). 



The star product is defined by (27) for the product of two planes or 

 complex two vectors. We can obtain Mo * Mi by interchanging A 

 with C and B with D. It follows from (27) that 



ilf 2 *Mi= - Ml * M2. (29) 



In particular if Mi = Mo = M, we get 



M*M = Q. (30) 



Representing J/3 in the s;yTnbolic form EF, by direct expansion we get 

 Mi-{M^ * Mz) = M3- (I/2 * Ml). (31) 



8. Star product of two planes. Suppose the two planes Mi 

 and M-i lie in a 3-space. They can be written 



Ml = AXB, M2= AXC 



where A is a unit vector and B and C are perpendicular to A. Then 



Ml *M2 = BXC. 



This is a plane vccior perpendicular io Mi and M2 and having a magni- 

 tude equal to the product of the magnitudes of Mi and il/2 aiid the sine 

 of the angle between them. In particular the product vanishes when and 

 only when Mi and il/2 lie in the same plane. 



If Ml and il/2 are not contained in three dimensions they always 

 lie in a 4-space. In that space we can always find two planes .1 X C 

 and B X D perpendicular to and cutting both. We can then choose 

 the vectors A, B, C, D such that 



Ml - A X B, Mo = CXD. 



Then since 



A-B = A-D = C-B = C-D = 0, 



Ml * M2= {A-C)BXD-\- (B-D) A X C. (32) 



