Search results
Results from the WOW.Com Content Network
A bivector is an oriented plane element and a trivector is an oriented volume element, in the same way that a vector is an oriented line element. Given vectors a, b and c, the product is a trivector with magnitude equal to the scalar triple product, i.e.
A 1-blade is a vector. Every vector is simple. A 2-blade is a simple bivector. Sums of 2-blades are also bivectors, but not always simple. A 2-blade may be expressed as the wedge product of two vectors a and b: . A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors a, b, and c:
Loosely speaking, the measure of a vector is a length, the measure of a bivector is an area, and the measure of a trivector is a volume. If a vector is factored directly into projective and rejective terms using the geometric product = (+), then it is not necessarily obvious that the rejection term, a product of vector and bivector is even a ...
[3] The scalars and vectors have their usual interpretation and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities of 3D vector calculus that are derived as a cross product, such as oriented area, oriented angle of rotation, torque, angular momentum and the magnetic ...
In multilinear algebra, a multivector, sometimes called Clifford number or multor, [1] is an element of the exterior algebra Λ(V) of a vector space V.This algebra is graded, associative and alternating, and consists of linear combinations of simple k-vectors [2] (also known as decomposable k-vectors [3] or k-blades) of the form
Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
The element e 123 is a trivector and the pseudoscalar for the geometry. ... The geometric algebra used is Cl n (R), n ≥ 3, the algebra of the real vector space R n.
A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T (v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between ...