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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 .
Cross product – also known as the "vector product", a binary operation on two vectors that results in another vector. The cross product of two vectors in 3-space is defined as the vector perpendicular to the plane determined by the two vectors whose magnitude is the product of the magnitudes of the two vectors and the sine of the angle ...
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.
There are two lists of mathematical identities related to vectors: Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
The logical operation and takes the place of multiplication. The outer product of two logical vectors (u i) and (v j) is given by the logical matrix () = (). This type of matrix is used in the study of binary relations, and is called a rectangular relation or a cross-vector. [12]
There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product [a] returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering.
It can be seen that both multiplication rules follow from the same Fano diagram by simply renaming the unit vectors, and changing the sense of the center unit vector. Considering all possible permutations of the basis there are 480 multiplication tables and so 480 cross products like this. [14]
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge. The divergence of a tensor field T {\displaystyle \mathbf {T} } of non-zero order k is written as div ( T ) = ∇ ⋅ T {\displaystyle \operatorname {div} (\mathbf {T} )=\nabla \cdot \mathbf {T} } , a contraction of a tensor field ...