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The rotation vector is an example of an axial vector. Axial vectors, or pseudovectors, are vectors with the special feature that their coordinates undergo a sign change relative to the usual vectors (also called "polar vectors") under inversion through the origin, reflection in a plane, or other orientation-reversing linear transformation. [18]
The fundamental difference is that GA provides a new product of vectors called the "geometric product". Elements of GA are graded multivectors: scalars are grade 0, usual vectors are grade 1, bivectors are grade 2 and the highest grade (3 in the 3D case) is traditionally called the pseudoscalar and designated .
The equations for x and y can be combined to give + = (+) [2] [3] or + = (). This equation shows that σ and τ are the real and imaginary parts of an analytic function of x+iy (with logarithmic branch points at the foci), which in turn proves (by appeal to the general theory of conformal mapping) (the Cauchy-Riemann equations) that these particular curves of σ and τ intersect at ...
Animation of the torsion and the corresponding rotation of the binormal vector. Let r be a space curve parametrized by arc length s and with the unit tangent vector T.If the curvature κ of r at a certain point is not zero then the principal normal vector and the binormal vector at that point are the unit vectors
(For example, for a position vector of length meters, if all Cartesian basis vectors are changed from meters in length to meters in length, the length of the position vector remains unchanged at meters, although the vector components will all increase by a factor of ). The scalar product of a vector and a covector is invariant, because one has ...
When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.
The three vectors spanning a parallelepiped have triple product equal to its volume. (However, beware that the direction of the arrows in this diagram are incorrect.) In exterior algebra and geometric algebra the exterior product of two vectors is a bivector, while the exterior product of three vectors is a trivector. A bivector is an oriented ...
Bivectors are quantities from geometric algebra, clifford algebra and the exterior algebra, which generalise the idea of vectors into two dimensions. As vectors are to lines, so are bivectors to planes. So every plane (in any dimension) can be associated with a bivector, and every simple bivector is associated with a plane. This makes them a ...