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Parallel plane segments with the same orientation and area corresponding to the same bivector a ∧ b. [1] In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is ...
This theory stipulated that all the laws of physics should take the same form in all coordinate systems – this led to the introduction of tensors. The tensor formalism also leads to a mathematically simpler presentation of physical laws. The inhomogeneous Maxwell equation leads to the continuity equation:
Since the vector term of the vector bivector product the name dot product is zero when the vector is perpendicular to the plane (bivector), and this vector, bivector "dot product" selects only the components that are in the plane, so in analogy to the vector-vector dot product this name itself is justified by more than the fact this is the non ...
Given a bivector r = r 1 + hr 2, the ellipse for which r 1 and r 2 are a pair of conjugate semi-diameters is called the directional ellipse of the bivector r. [4]: 436 In the standard linear representation of biquaternions as 2 × 2 complex matrices acting on the complex plane with basis {1, h},
Specifically, a k-blade is a k-vector that can be expressed as the exterior product (informally wedge product) of 1-vectors, and is of grade k. In detail: [1] A 0-blade is a scalar. 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 multivector that is the exterior product of linearly independent vectors is called a blade, and is said to be of grade . [f] A multivector that is the sum of blades of grade is called a (homogeneous) multivector of grade . From the axioms, with closure, every multivector of the geometric algebra is a sum of blades.