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.


1 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.

  • $\begingroup$ Thank you very much I understood it better, I only had the doubt about internal orientation and outer orientation. $\endgroup$ 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
    Apr 21, 2019 at 21:34

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