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The geometric algebra is Cl 3,1 (R), and the subspace of bivectors is ⋀ 2 R 3,1. The simple bivectors are of two types. The simple bivectors e 23, e 31 and e 12 have negative squares and span the bivectors of the three-dimensional subspace corresponding to Euclidean space, R 3. These bivectors generate ordinary rotations in R 3.
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A two-vector or bivector [1] is a tensor of type () and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of ...
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 ...
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:
In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors.Geometric algebra is built out of two fundamental operations, addition and the geometric product.
The electromagnetic field at a point p (i.e. an event) of a Lorentzian spacetime is represented by a real bivector F = F ab defined over the tangent space at p. The tangent space at p is isometric as a real inner product space to E 1,3. That is, it has the same notion of vector magnitude and angle as Minkowski spacetime.
The narrative underlines the authority of Peter, who could see through the deception by Ananias and Sapphira (verses 3–5, 8–9) and highlights the spiritual authority of the "church" (Greek: ekklesia, first used in Acts in verse 11) in form of 'signs' of God (inducing 'great fear' in verses 5 and 11, as well as healing miracles in the next section). [6]