VECTOR ANALYSIS. 89 



component is the same in both, but the major axes of the ellipses are 

 perpendicular. The case in which the directions of t and $ are real, 

 forms no exception to this ride. 



It will be observed that every circular bi vector is perpendicular to 

 itself, and to every parallel bivector. 



If two bivectors, JUL + IV, /UL' + IV', which do not lie in the same plane 

 are perpendicular, we may resolve /UL and v into components parallel 

 and perpendicular to the plane of /z' and i/. The components perpen- 

 dicular to the plane evidently contribute nothing to the value of 



[/X + <!/]. [// + >']. 



Therefore the components of /u. and v parallel to the plane of JUL', v, 

 form a bivector which is perpendicular to /UL'+IV. That is, if two 

 bivectors are perpendicular, the directional ellipse of either, projected 

 upon the plane of the other and rotated through a quadrant in that 

 plane, will be similar and similarly situated to the directional ellipse 

 of the second. 



8. A bivector may be divided in one and only one way into parts 

 parallel and perpendicular to another, provided that the second is not 

 circular. If a and b are the bivectors, the parts of a will be 



b.a t -, b.a t 

 r-r b and a ^-r b. 

 b.b b.b 



If b is circular, the resolution of a is impossible, unless it is perpen- 

 dicular to b. In this case the resolution is indeterminate. 



9. Since axb.a = 0, and axb.b = 0, axb is perpendicular to a and b. 

 We may regard the plane of the product as determined by the 

 condition that the directional ellipses of the factors projected upon 

 it become similar and similarly situated. The directional ellipse of 

 the product is similar to these projections, but its orientation is 

 different by 90. It may easily be shown that axb vanishes only 

 with a or b, or when a and b are parallel. 



10. The bivector equation 



(axb.c)b (b.cxb)a+(c.bxa)b (b.axb)c = 



is identical, as may be verified by substituting expressions of the 

 form xi + yj+zk (x, y, z being biscalars), for each of the bivectors. 

 (Compare No. 37.) This equation shows that if the product axb 

 of any two bivectors vanishes, one of these will be equal to the other 

 with a biscalar coefficient, that is, they will be parallel, according 

 to the definition given above. If the product a.bxc of any three 

 bivectors vanishes, the equation shows that one of these may be 

 expressed as a sum of the other two with biscalar coefficients. In 

 this case, we may say (from the analogy of the scalar analysis) that 

 the three bivectors are complanar. (This does not imply that they 



