state the definations and representations of vectors?

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    In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector has properties that do not depend on the coordinate system used to describe it. However, a vector is often described by a fixed number of components, each of which is dependent upon the particular coordinate system being used, such as Cartesian coordinates, spherical coordinates or polar coordinates.

    A common example of a vector is force — it has a magnitude and an orientation and multiple forces sum according to the parallelogram law.

    A spatial vector can be formally defined by its relationship to the spatial coordinate system under rotations. Alternatively, it can be defined in a coordinate-free fashion via a tangent space of a three-dimensional manifold in the language of differential geometry. These definitions are discussed in more detail below.

    A spatial vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry). Vectors are the building blocks of vector fields and vector calculus.

    Informally, a vector is a quantity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction, often represented graphically by an arrow. Examples are "moving north at 90 km/h" or "pulling towards the center of Earth with a force of 70 newtons".

    The notion of having a "magnitude" and "direction" is formalized by saying that the vector has components that transform like the coordinates under rotations. That is, if the coordinate system undergoes a rotation described by a rotation matrix R, so that a coordinate vector x is transformed to x′ = Rx, then any other vector v is similarly transformed via v′ = Rv. This ensures the invariance of the operations dot product, Euclidean norm, cross product, gradient, divergence, curl, and scalar triple product, and trivially for vector addition and subtraction, and scalar multiplication. The terms scalar and vector as used here include pseudoscalars and pseudovectors or axial vectors (see also below).

    In mathematics, a vector is considered more than a representation of a physical quantity. In general, a vector is any element of a vector space over some field. The spatial vectors of this article are a very special case of this general definition (they are not simply any element of Rd in d dimensions), which includes a variety of mathematical objects (algebras, the set of all functions from a given domain to a given linear range, and linear transformations). Note that under this definition, a tensor is a special vector!

    Vectors are usually denoted in boldface, as a. Other conventions include \vec{a} or a, especially in handwriting. Alternately, some use a tilde (~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type.

    Vectors are usually shown in graphs or other diagrams as arrows

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