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In exterior algebra and geometric algebra the exterior product of two vectors is a bivector, while the exterior product of three vectors is a trivector. 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
If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). [4] One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple.
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.
Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relation is formally defined as a set of pairs , i.e. a subset of the Cartesian product A × B of some sets A and B , so a ternary relation is a set of triples, forming a subset of the Cartesian product A × B × C of three sets A , B and C .
Let K 3 denote the set of all triples x = (x 0, x 1, x 2) of elements of K (a Cartesian product viewed as a vector space). For any nonzero x in K 3, the minimal subspace of K 3 containing x (which may be visualized as all the vectors in a line through the origin) is the subset {:} of K 3.
3 Two sets involved. ... and binary Cartesian product ... which is a binary operation, has the following three properties: [3] ...
In the Cartesian plane, two vectors are said to be perpendicular if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero.
The lowest-dimensional forms are 6-polytopes being the Cartesian product of three polygons. The smallest can be written as {3} × {3} × {3} in Schläfli symbols if they are regular, and contains 27 vertices. This is the product of three equilateral triangles and is a uniform polytope. The f-vectors can be computed by (3,3,1) 3 = (27,81,108,81 ...