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These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Vectors play an important role in physics: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. [7]
These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Vectors play an important role in physics: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. [7]
Aside from the notion of units and support, physical vector quantities may also differ from Euclidean vectors in terms of metric. For example, an event in spacetime may be represented as a position four-vector, with coherent derived unit of meters: it includes a position Euclidean vector and a timelike component, t ⋅ c 0 (involving the speed ...
In physics, a vector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a covector.
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied ("scaled") by numbers called scalars. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms .
(For example, for a position vector of length meters, if all Cartesian basis vectors are changed from meters in length to meters in length, the length of the position vector remains unchanged at meters, although the vector components will all increase by a factor of ). The scalar product of a vector and a covector is invariant, because one has ...
For example, the angular momentum is a pseudovector because it is often described as a vector, but by just changing the position of reference (and changing the position vector), angular momentum can reverse direction, which is not supposed to happen with true vectors (also known as polar vectors). [3] One example of a pseudovector is the normal ...
For example, the complex numbers C form a two-dimensional vector space over the real numbers R. Likewise, the real numbers R form a vector space over the rational numbers Q which has (uncountably) infinite dimension, if a Hamel basis exists. [b] If V is a vector space over F it may also be regarded as vector space over K. The dimensions are ...