21 Facts About Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector or simply a vector is a geometric object that has magnitude (or length) and direction.

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 Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity.

The mathematical representation of a physical Euclidean vector depends on the coordinate system used to describe it.

Term Euclidean vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum of a Real number and a 3-dimensional Euclidean vector.

In physics and engineering, a Euclidean vector is typically regarded as a geometric entity characterized by a magnitude and a direction.

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In pure mathematics, a Euclidean vector is defined more generally as any element of a Euclidean vector space.

Thus two arrows and in space represent the same free Euclidean vector if they have the same magnitude and direction: that is, they are equipollent if the quadrilateral ABB'A' is a parallelogram.

Term Euclidean vector has generalizations to higher dimensions, and to more formal approaches with much wider applications.

Euclidean vector space is often presented as the Euclidean vector space of dimension.

Examples of quantities that have magnitude and direction, but fail to follow the rules of Euclidean vector addition, are angular displacement and electric current.

For instance, the points and in space determine the bound Euclidean vector pointing from the point on the x-axis to the point on the y-axis.

Thus the free Euclidean vector represented by is a Euclidean vector of unit length—pointing along the direction of the positive x-axis.

The Euclidean vector itself has not changed, but the basis has, so the components of the Euclidean vector must change to compensate.

The Euclidean vector is called covariant or contravariant, depending on how the transformation of the Euclidean vector's components is related to the transformation of the basis.

The endpoint of a Euclidean vector can be identified with an ordered list of n real numbers.

Notion that the tail of the Euclidean vector coincides with the origin is implicit and easily understood.

The Euclidean vector is said to be decomposed or resolved with respect to that set.

Unlike any other Euclidean vector, it has an arbitrary or indeterminate direction, and cannot be normalized.

However, a Euclidean vector can be expressed in terms of any number of different bases that are not necessarily aligned with each other, and still remain the same Euclidean vector.

In other words, if the reference axes were rotated in one direction, the component representation of the Euclidean vector would rotate in the opposite way to generate the same final Euclidean vector.

Mathematically, if the basis undergoes a transformation described by an invertible matrix M, so that a coordinate vector x is transformed to, then a contravariant vector v must be similarly transformed via v This important requirement is what distinguishes a contravariant vector from any other triple of physically meaningful quantities.