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A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". [4] It was first used by 18th century astronomers investigating planetary revolution around the Sun. [5] The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B.
A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment (A, B)) and same direction (e.g., the direction from A to B). [14] In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific ...
The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
Such a vector consists of a magnitude (or length) and a direction (or angle). The magnitude, typically represented as r, is the distance from a starting point, the origin, to the point which is represented.
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...
A direction in (n + 1)-dimensional space will be a unit magnitude vector, which we may consider a point on a generalized sphere, S n. Thus it is natural to describe the rotation group SO(n + 1) as combining SO(n) and S n. A suitable formalism is the fiber bundle, (+),
The curl of a 3-dimensional vector field which only depends on 2 coordinates (say x and y) is simply a vertical vector field (in the z direction) whose magnitude is the curl of the 2-dimensional vector field, as in the examples on this page.
By definition, all Euclidean vectors have a magnitude (see above). However, a vector in an abstract vector space does not possess a magnitude. A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space. [8] The norm of a vector v in a normed vector space can be considered to be the magnitude of v.