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In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions n and m , then their outer product is an n × m matrix.
The exterior algebra is named after Hermann Grassmann, [3] and the names of the product come from the "wedge" symbol and the fact that the product of two elements of is "outside" . The wedge product of k {\displaystyle k} vectors v 1 ∧ v 2 ∧ ⋯ ∧ v k {\displaystyle v_{1}\wedge v_{2}\wedge \dots \wedge v_{k}} is called a blade of degree k ...
As an example, the geometric product of two vectors = + = + since = and = and = , for other than and . A multivector A {\displaystyle A} may also be decomposed into even and odd components, which may respectively be expressed as the sum of the even and the sum of the odd grade components above:
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.
The cross product (also called the vector product or outer product) is only meaningful in three or seven dimensions. The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector. The cross product, denoted a × b, is a vector perpendicular to both a and b and is defined as
The cross product with respect to a right-handed coordinate system. In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol .
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The divergence of a second-order tensor field in cylindrical polar coordinates can be obtained from the expression for the gradient by collecting terms where the scalar product of the two outer vectors in the dyadic products is nonzero.