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.

  • 2
    $\begingroup$ I suspect the more common names for what you call "axial vector" and "polar vector" are "pseudovector" and "vector". See this answer of mine for another explanation of the difference between them. $\endgroup$
    – ACuriousMind
    Apr 19 '19 at 20:51
  • $\begingroup$ Possible duplicate. At least worth reading the answers therein. $\endgroup$
    – jacob1729
    Apr 19 '19 at 21:07

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.


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