Likewise, the momentum vector is the velocity vector ( a polar vector ) times mass ( a scalar ), so is a polar vector.
12.
Angular momentum is the cross product of a displacement ( a polar vector ) and momentum ( a polar vector ), and is therefore a pseudovector.
13.
Angular momentum is the cross product of a displacement ( a polar vector ) and momentum ( a polar vector ), and is therefore a pseudovector.
14.
They better disambiguate axial and polar vectors, that is the quantities represented by them, so it is clearer which operations are allowed and what their results are.
15.
Relevant is the distinction between axial and polar vectors in vector algebra, which is natural in geometric algebra as the distinction between vectors and bivectors ( elements of grade two ).
16.
This behavior of cross products is consistent with their definition as vector-like elements that change sign under transformation from a right-handed to a left-handed coordinate system, unlike polar vectors.
17.
The definition of a " vector " in physics ( including both polar vectors and pseudovectors ) is more specific than the mathematical definition of " vector " ( namely, any element of an abstract vector space ).
18.
A polar vector can be represented in calculations ( for example, when rotated by a quaternion " similarity transform " ) by a pure imaginary quaternion, with no loss of information, but the two should not be confused.
19.
For example, the inner product of a polar vector and an axial vector resulting from the cross product in the triple product should result in a pseudoscalar, a result which is more obvious if the calculation is framed as the exterior product of a vector and bivector.
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