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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.
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.
In mathematics and physics, vector notation is a commonly used notation for representing vectors, [1] [2] which may be Euclidean vectors, or more generally, members of a vector space. For denoting a vector, the common typographic convention is lower case, upright boldface type, as in v.
The vector projection of a on b is a vector a 1 which is either null or parallel to b. More exactly: a 1 = 0 if θ = 90°, a 1 and b have the same direction if 0° ≤ θ < 90°, a 1 and b have opposite directions if 90° < θ ≤ 180°.
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
A vector is a quantity that has both magnitude and direction. Represented as A = (a₁, a₂, a₃) in 3D space. Notation: Bold letters (A), A with an arrow (), or component form. 2. Types of Vectors • Zero Vector (\mathbf{0}): Magnitude is zero. • Unit Vector (\hat{A}): Magnitude is one. • Equal Vectors: Same magnitude and direction.
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.
A bound vector is defined as the combination of an ordinary vector quantity and a point of application or point of action. [1] [4] Bound vector quantities are formulated as a directed line segment, with a definite initial point besides the magnitude and direction of the main vector. [1] [3] For example, a force on the Euclidean plane has two ...