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The length of a vector is defined as the square root of the dot product of the vector by itself, and the cosine of the (non oriented) angle between two vectors of length one is defined as their dot product. So the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations of ...
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.
In a geometric algebra for which the square of any nonzero vector is positive, the inner product of two vectors can be identified with the dot product of standard vector algebra. The exterior product of two vectors can be identified with the signed area enclosed by a parallelogram the sides of which are the vectors.
1698 (perhaps deriving from a much earlier use of middle dot to separate juxtaposed numbers) division slash (a.k.a. solidus ) 1718 (deriving from horizontal fraction bar, invented by Abu Bakr al-Hassar in the 12th century)
The generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connection, which differentiates a vector field to give a vector-valued 1-form. Cross product rule
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In terms of the wedge product, Lagrange's identity can be written () = ().. Hence, it can be seen as a formula which gives the length of the wedge product of two vectors, which is the area of the parallelogram they define, in terms of the dot products of the two vectors, as ‖ ‖ = () = ‖ ‖ ‖ ‖ ().