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In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [9] [10] It is typically formulated as the product of a unit of measurement and a vector numerical value (), often a Euclidean vector with magnitude and direction.
A map from this sphere to a unit sphere of dimension n − 1 can be constructed by dividing each vector on this sphere by its length to form a unit length vector, which is a point on the unit sphere S n−1. This defines a continuous map from S to S n−1. The index of the vector field at the point is the degree of this map. It can be shown ...
Most notably, in a 1929 paper he set out an argument based on the Pauli exclusion principle and the Dirac equation that fixed the value of the reciprocal of the fine-structure constant as 𝛼 −1 = 16 + 1 / 2 × 16 × (16–1) = 136. When its value was discovered to be closer to 137, he changed his argument to match that value.
A scalar in physics and other areas of science is also a scalar in mathematics, as an element of a mathematical field used to define a vector space.For example, the magnitude (or length) of an electric field vector is calculated as the square root of its absolute square (the inner product of the electric field with itself); so, the inner product's result is an element of the mathematical field ...
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 ...
For example, the speed of light is defined as having the numerical value of 299 792 458 when expressed in the SI unit metres per second, and as having the numerical value of 1 when expressed in the natural units Planck length per Planck time. While its numerical value can be defined at will by the choice of units, the speed of light itself is a ...
[1] [2] The term normalized vector is sometimes used as a synonym for unit vector. A unit vector is often used to represent directions , such as normal directions . Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination form of unit vectors.
The tangent bundle of the circle S 1 is globally isomorphic to S 1 × R, since there is a global nonzero vector field on S 1. [nb 12] In contrast, by the hairy ball theorem, there is no (tangent) vector field on the 2-sphere S 2 which is everywhere nonzero. [92] K-theory studies the isomorphism classes of all vector bundles over some ...