Vector, axial orthogonal

Fig. 2. A simple biomechanical analysis looking at the resolution of a single axial compressive force vector into orthogonal components at the L3-4, L4-5, and L5-S1 levels. The analysis demonstrates the increased anterior shear forces at the lower lumbar levels, particularly the lumbosacral level. The analysis is overly simplistic in that it does not include the effect of the posterior musculature which may counteract these large anterior shear forces...

The vector or cross product determines a so-called axial vector with orthogonal orientation from two spatial vectors. The vector product is anti-commutative ... 

Therefore the held results from the addition of two distinct helds a polar held and an axial held . In the available literature, the expressions longitudinal and transverse, which respectively refer to the and vectors, do not necessarily imply that these vectors are projections of on orthogonal direchons. Indeed (if we leave Beltrami helds apart that fulfill the condition A VA 0), for A VA A = 0, the vector B = VA A is normal to if A is parallel to . If not, that is, when A L , both and have the same direchon. This analysis demonstrates the V A VA operator, as applied to the vector A, is itself a vector having either the same direction as A or having components in direchons parallel to A and perpendicular to the plan originating from vectors A and V A A. 

In metals, where there is only one type of charge carrier, the Hall coefficient, Rb = (Ey / jxB), is very useful for measuring both the carrier density and the magnetic field B. Since the secondary field Ey and B are orthogonal, the Hall effect is a transverse effect and the Hall tensor is of rank three. It relates the axial vector Bj. to the antisymme-trical second-rank tensor Ey, which is equivalent to p yjx, where p y = PxyzBz-... 

As distinct from usual polar vectors which by coordinate changes (characterized by orthogonal matrix , see Rem. 8) transform by (b) of Rem. 4, the axial vector w is defined by transformation... 

In the exact theory of P decay the selection rules are derived from the transformation properties of operators and orthogonality of wave functions of different angular momentum states. For the actual vector and axial vector interactions the following selection rules have been obtained ... 

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