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For example, in six dimensions with standard basis (e 1, e 2, e 3, e 4, e 5, e 6) the bivector + + is the sum of three simple bivectors but no less. As in four dimensions it is always possible to find orthogonal simple bivectors for this sum.
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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 ...
The torque or curl is then a normal vector field in this 3rd dimension. By contrast, geometric algebra in 2 dimensions defines these as a pseudoscalar field (a bivector), without requiring a 3rd dimension. Similarly, the scalar triple product is ad hoc, and can instead be expressed uniformly using the exterior product and the geometric product.
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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:
The chapter begins with God giving Moses a message for the Israelites about the need to be holy, to respect parents, and to avoid idolatry (verses 1–4). Next are instructions for peace offerings (5-8), food aid for poor people and foreigners (9-10), and various instructions relating to ethical treatment of others (11-18) and agricultural ...
Author: Orr, James, 1844-1913, ed: Short title: The International standard Bible encyclopedia; Date and time of digitizing: 03:09, 24 November 2009: Software used