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Axial Vectors in Rotating Coordinate Systems

Axial vectors are vector cross products of ordinary position vectors.
For example, angular momentum L=r×v and torque T=r×F are
axial vectors.

Note that in the following the typographically distinction
between vectors and scalars is that vectors are shown in color.
Ordinary vectors are shown in red
and axial vectors in green. Sometimes axial vectors are called
pseudovectors
and ordinary position vectors are called polar vectors.
The significant difference between axial vectors and polar vectors
is the
effect on their coordinates of an inversion of the coordinate system;
i.e.,

x → -x

y → -y

z → -z

Under this inversion an ordinary (polar) vector
P is transformed into its
negative −P, whereas the coordinates of an axial vector A are unchanged
by the inversion; i.e.,

P → -P

A → A

The time derivative of a polar vector can be represented by the
following formula

dP/dt = d_{r}P/dt + Ω×P

where dP/dt is the time derivative with respect to coordinates in
an intertial frame of reference whereas d_{r}P/dt is the
time derivative with respect to a frame of reference which is rotating
about the vector Ω at a rate of rotation of Ω.

Now consider the axial vector A=B×C.
Its time rate of change
with respect to an inertial frame of reference is given by:

When these expressions are substituted into the formula for dA/dt
two terms cancel out leaving

dA/dt = d_{r}A/dt + (C^{.}Ω)B - (B^{.}Ω)C

When the above expression is rearranged to

dA/dt = d_{r}A/dt + (Ω^{.}C)B - (Ω^{.}B)C

the Lagrange triple vector product formula can be applied in reverse
to yield

dA/dt = d_{r}A/dt + Ω×(B×C)

This shows that the formula for the time derivative of a vector under
a rotating frame of reference applies to axial vectors as well as
ordinary position vectors. It would also apply to hybrid vectors
such as a vector cross product of a postion vector and an axial vector
or the vector cross product of axial vectors.