Search results
Results from the WOW.Com Content Network
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.
In a d-dimensional space, Hodge star takes a k-vector to a (d–k)-vector; thus only in d = 3 dimensions is the result an element of dimension one (3–2 = 1), i.e. a vector. For example, in d = 4 dimensions, the cross product of two vectors has dimension 4–2 = 2, giving a bivector. Thus, only in three dimensions does cross product define an ...
By referring collectively to e 1, e 2, e 3 as the e basis and to n 1, n 2, n 3 as the n basis, the matrix containing all the c jk is known as the "transformation matrix from e to n", or the "rotation matrix from e to n" (because it can be imagined as the "rotation" of a vector from one basis to another), or the "direction cosine matrix from e ...
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 dot product in Cartesian coordinates (Euclidean space with an orthonormal basis set) is simply the sum of the products of components. In orthogonal coordinates, the dot product of two vectors x and y takes this familiar form when the components of the vectors are calculated in the normalized basis:
Every finite-dimensional inner product space is also a Hilbert space. [2] The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ‖ x ‖, and to the angle θ between two vectors x and y by means of the formula = ‖ ‖ ‖ ‖ .
where {e 1 ∧ e 2, e 3 ∧ e 1, e 2 ∧ e 3} is the basis for the three-dimensional space ⋀ 2 (R 3). The coefficients above are the same as those in the usual definition of the cross product of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a bivector. Bringing in a ...