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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.

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  • $\begingroup$ The terminology in this quote is the wrong way round: Usually axial vectors are said to have an outer orientation that depends on the "surrounding" space. They are often depicted by a line segment with a small arrow bent around its center. They might be thought of as turning around their own axis, hence "axial". Polar vectors otoh are the "usual" vectors depicted as straight arrows and are said to have an inner orientation. They point somewhere, maybe to a pole, hence "polar". See for instance: Weinreich Geometrical Vectors $\endgroup$
    – Michael Bächtold
    Commented Nov 14 at 12:31

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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.

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  • $\begingroup$ Thank you very much I understood it better, I only had the doubt about internal orientation and outer orientation. $\endgroup$
    – Daniel OnoRoc
    Commented Apr 21, 2019 at 16:20
  • $\begingroup$ It is a subtle point, and probably unimportant for your learning. If a force vector has the coordinates changed from right-hand to left-hand, it still remains pointed along the axis (axial vector) of the force. However, a torque is the dot product of a force times a directional unit vector, so if the unit vector changes direction, the polar vector, or torque, changes direction also. It is more notational than physical. $\endgroup$
    – SteveO
    Commented Apr 21, 2019 at 21:34

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