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Parallelepiped, generated by three vectors A parallelepiped is a prism with a parallelogram as base. Hence the volume V {\displaystyle V} of a parallelepiped is the product of the base area B {\displaystyle B} and the height h {\displaystyle h} (see diagram).
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 .
The vector area of a parallelogram is given by the cross product of the two vectors that span it; it is twice the (vector) area of the triangle formed by the same vectors. In general, the vector area of any surface whose boundary consists of a sequence of straight line segments (analogous to a polygon in two dimensions) can be calculated using ...
The selection of an origin and a basis define a unit cell, a parallelotope (i.e., generalization of a parallelogram (2D) or parallelepiped (3D) in higher dimensions) defined by the lattice basis vectors ,, …, where is the dimension of the space.
The three vectors spanning a parallelepiped have triple product equal to its volume. (However, beware that the direction of the arrows in this diagram are incorrect.) 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 ...
The other coordinates can be obtained from vector addition [5] of the 3 direction vectors: e 1 + e 2, e 1 + e 3, e 2 + e 3, and e 1 + e 2 + e 3. The volume V {\displaystyle V} of a rhombohedron, in terms of its side length a {\displaystyle a} and its rhombic acute angle θ {\displaystyle \theta ~} , is a simplification of the volume of a ...
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 fundamental difference is that GA provides a new product of vectors called the "geometric product". Elements of GA are graded multivectors: scalars are grade 0, usual vectors are grade 1, bivectors are grade 2 and the highest grade (3 in the 3D case) is traditionally called the pseudoscalar and designated .