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The 6-sphere, or hypersphere in seven dimensions, is the six-dimensional surface equidistant from a point. It has symbol S 6, and the equation for the 6-sphere, radius r, centre the origin is = {: ‖ ‖ =}. The volume of the space bounded by this 6-sphere is
A diagram of dimensions 1, 2, 3, and 4. In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. [1] [2] It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension.
The term vector was coined by W. R. Hamilton around 1843, as he revealed quaternions, a system which uses vectors and scalars to span a four-dimensional space. For a quaternion q = a + b i + c j + d k, Hamilton used two projections: S q = a , for the scalar part of q , and V q = b i + c j + d k, the vector part.
In 3 dimensions the curl of a vector field is a vector field as is familiar (in 1 and 0 dimensions the curl of a vector field is 0, because there are no non-trivial 2-vectors), while in 4 dimensions the curl of a vector field is, geometrically, at each point an element of the 6-dimensional Lie algebra ().
By Dvoretzky's theorem, every finite-dimensional normed vector space has a high-dimensional subspace on which the norm is approximately Euclidean; the Euclidean norm is the only norm with this property. [24] It can be extended to infinite-dimensional vector spaces as the L 2 norm or L 2 distance. [25]
A vector pointing from A to B. In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector [1] or spatial vector [2]) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space.
Linear subspace of dimension 1 and 2 are referred to as a line (also vector line), and a plane respectively. If W is an n-dimensional vector space, any subspace of dimension 1 less, i.e., of dimension is called a hyperplane. [53] The counterpart to subspaces are quotient vector spaces. [54]
For a three-dimensional inner product space, the notions of projection of a vector onto another and rejection of a vector from another can be generalized to the notions of projection of a vector onto a plane, and rejection of a vector from a plane. [5] The projection of a vector on a plane is its orthogonal projection on that plane. The ...