What is an axial and polar vector?

Question

Can someone please explain this type of vector to me, I can not understand it.

• Axial vectors have an inner orientation, i.e. the direction of the vector indicates the positive orientation. For example, a unit linear force vector: the positive direction of the force does not depend on the orientation (right-handed vs. left-handed) of the world reference frame. As many (but not all) other textbooks, this book implicitly uses right-handed reference frames only, but no physical arguments prevent the use of left-handed frames.

• Polar vectors have an outer orientation, i.e. the positive orientation cannot be derived from the direction vector itself, but is imposed on it by the environment." For example, a unit moment of force vector: if the handedness of the world frame changes, the orientation associated with the moment vector changes too. Note that this is a feature of the coordinate representation, not of the physical property that the vector stands for.

Answer 1

A polar vector can be described by one unit vector, $V^i e_i$. Where $e_i$ is a unit vector. If you use another unit vector $f_i$ such that $f_i = -e_i$ then the sign of the vector changes. Example velocity

A bi-vector can be described by two unit vectors. The two unit vectors form a small patch of area. The area contain orientation i.e. you can choose to form the area from unit vector 1 to 2 or from unit vector 2 to 1. So , bi-vectors have anti symmetric unit vectors. (study the wedge products of the two unit vectors.). Now if you change you unit vectors like this $f_i = -e_i$ , the sign of the bi-vectors does not change because the two negative signs of the unit vectors cancel out.

Axial vector, in 3 dimension, you can assign a vector to any patch of area. Now if you assign a vector to patch of the area which is created by wedge product of two vectors, that vector is called axial vector. It is important because if you change you unit vectors like this $f_i = -e_i$ the axial vector do not change sign (unlike the polar vectors) Examples , torque, magnetic field, and angular momentum.

Answer 2

I suppose that these notions are dealing with the property of inverse coordinate transformation which is known as parity transformation.

Polar Vector is called true vector, contravariant vector which represents true physical quantities in any coordinate system.

Axial vector is called pseudovector since its use for the real physical inputs can generate artificial (even wrong, i.e., reflected and reversed) output.

The components of a polar vector change signs during inversion, while the axial vector does not change sign of its components.

Note here the sign change in the component does not mean the vector change direction since the sign change of (reciprocal) basis vector affects the product.

However, the vector product of either two polar vectors or two axial ones will be an axial vector and the vector product of one polar vector and one axial vector will become a polar vector.

Inversion for polar vectors has the similar property as involutions (negation).

Answer 3

Axial vectors depend upon the orientation of a vector space in that the change to the opposite direction if the orientation is changed to the opposite orientation. However, I don't think this can be made sense of as orientation is not a property of vectors but the entire vector space. Instead, we define an orientation dependent bilinear map on a vector space $V$ and valued in a vector space $W$ as:

$b : V \times V \times Or(V) \rightarrow W $

Here $Or(V)$ is the set of all orientations on $V$. And we say it is ordinary when:

$b(u,v, -o) = b(u,v, o)$

for all $u,v\in V$ and all $o \in Or(V)$. This is the same as an ordinary bilinear map. We say it is axial when:

$b(u,v, -o) = -$b(u,v, o)

for all $u,v\in V$ and all $o \in Or(V)$. Then we discover the cross product is an axial bilinear map. This is how axiality is encoded. Thus axiality is not a property of vectors but a property of a multiplication.