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The full geometric algebra in three dimensions, Cl 3 (R), has basis (1, e 1, e 2, e 3, e 23, e 31, e 12, e 123). The element e 123 is a trivector and the pseudoscalar for the geometry. Bivectors in three dimensions are sometimes identified with pseudovectors [ 17 ] to which they are related, as discussed below .
Therefore, every blade of grade can be written as the exterior product of vectors. More generally, if a degenerate geometric algebra is allowed, then the orthogonal matrix is replaced by a block matrix that is orthogonal in the nondegenerate block, and the diagonal matrix has zero-valued entries along the degenerate dimensions.
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 a vector space of dimension n, a blade of grade n − 1 is called a pseudovector [2] or an antivector. [3] The highest grade element in a space is called ...
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 following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [9] [10] It is typically formulated as the product of a unit of measurement and a vector numerical value (), often a Euclidean vector with magnitude and direction.
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors.The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.
Pentagram - star polygon with 5 sides; Hexagram – star polygon with 6 sides Star of David (example) Heptagram – star polygon with 7 sides; Octagram – star polygon with 8 sides Star of Lakshmi (example) Enneagram - star polygon with 9 sides; Decagram - star polygon with 10 sides; Hendecagram - star polygon with 11 sides; Dodecagram - star ...