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|Now it is known that a scalar is a quantity which is invariant under any change of |
axes while a pseudoscalar is a quantity which changes sign when the axes
change ... The scalar product of two such vectors again gives a scalar. Hence,
|In even dimensions, the pseudoscalar anticommutes with all vectors, as we have |
already seen in two dimensions. We can now express each of our basis bivectors
as the product of the pseudoscalar and a dual vector: 2 = /e3, e2e3 = 7ei, ...
|Under the assumption that v is real, v (£ Z, we can now build a pseudo-scalar |
product on 2l„, invariant under the z/-anaplectic representation. We shall consider
the case when v g] — 1, 0[+2Z, in which, as will be seen, it is possible to build a ...
|left or the right, the product of a unit vector ei (i = 1, 2, or 3) with the unit grade-3 |
right-handed pseudoscalar I returns a bivector which represents the plane
perpendicular to the original vector. This operation of multiplication is called the
|But not each number is a scalar: For example the x component of a vector is a |
number, but no scalar, because it changes ... The cross product of two polar
vectors (e.g., in the case of the angular momentum) is an axial vector (pseudo
|There are scalars and vectors known as pseudo scalars and pseudo or axial |
vectors; there have transformation rules that involve a change in sign when ... An
example of a pseudo vector is the vector product u >< v of two polar vectors a and
|The cross product of the two polar vectors vvv and uuu is achiral (pseudo) vector |
ΩΩΩ = vvv×uuu. Think of the right hand rule for the direction of ΩΩΩ which
reverses on use of the left hand. course, a product of two pseudo-scalars is a true
|A pseudoscalar quantity is a number with no directional properties but which |
changes sign under P. A pseudoscalar is generated by taking the scalar product
of a polar and an axial vector. Physical quantities are classified as time-even or ...
|These products also represent the dual relationships between vectors and |
bivectors in 3D. Table 7.11 Inner product of the 3D basis vectors and the
pseudoscalar Inner product I123 e1 e23 e2 e31 e3 e12 Table 7.12 Reverse inner
product of ...
|It is common to normalize this to a unit pseudoscalar and to denote it by In or In. |
The choice of the sign of the unit ... The familiar 3D cross product of vectors can
be made in CGA as x× a = (x∧ a)I−13, though its use should be avoided. Duality
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