What is an axial and polar vector?

Question

In Robot Kinematics and Dynamics by Herman Bruyninckx, it states:

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 vector 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.

Can someone please explain these types of vector to me? I do not understand the explanation given.

Answer 1

It may be helpful to not get caught up in the author’s description of frames with respect to these vectors. The author is stating that an axial vector, like a force vector, acts along a line. But a polar vector, like a torque, acts about an axis.