Vector quantity

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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. [1] [2] It is typically formulated as the product of a unit of measurement and a vector numerical value (unitless), often a Euclidean vector with magnitude and direction. For example, a position vector in physical space may be expressed as three Cartesian coordinates with SI unit of meters.

In physics and engineering, particularly in mechanics, a physical vector may be endowed with additional structure compared to a geometrical vector. [3] 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 Cartesian components in SI unit of newtons and an accompanying two-dimensional position vector in meters, for a total of four numbers on the plane (and six in space). [5] [6] [4] A simpler example of a bound vector is the translation vector from an initial point to an end point; in this case, the bound vector is an ordered pair of points in the same position space, with all coordinates having the same quantity dimension and unit (length an meters). [7] [8] A sliding vector is the combination of an ordinary vector quantity and a line of application or line of action, over which the vector quantity can be translated (without rotations). A free vector is a vector quantity having an undefined support or region of application; it can be freely translated with no consequences; a displacement vector is a prototypical example of free vector.

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 ⋅ c0 (involving the speed of light). In that case, the Minkowski metric is adopted instead of the Euclidean metric.

Vector quantities are a generalization of scalar quantities and can be further generalized as tensor quantities. [8] Individual vectors may be ordered in a sequence over time (a time series), such as position vectors discretizing a trajectory. A vector may also result from the evaluation, at a particular instant, of a continuous vector-valued function (e.g., the pendulum equation). In the natural sciences, the term "vector quantity" also encompasses vector fields defined over a two- or three-dimensional region of space, such as wind velocity over Earth's surface. Pseudo vectors and bivectors are also admitted as physical vector quantities.

See also

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<span class="mw-page-title-main">Euclidean vector</span> Geometric object that has length and direction

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In mathematics, the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an ordering of the class of objects to which it belongs. Magnitude as a concept dates to Ancient Greece and has been applied as a measure of distance from one object to another. For numbers, the absolute value of a number is commonly applied as the measure of units between a number and zero.

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<span class="mw-page-title-main">Three-dimensional space</span> Geometric model of the physical space

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Scalars are physical quantities that are unaffected by changes to a vector space basis. Scalars are often accompanied by units of measurement, as in "10 cm". Examples of scalar quantities are mass, distance, charge, volume, time, speed, and the magnitude of physical vectors in general.

The mathematics of general relativity is complicated. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solved by algebra alone. In relativity, however, an object's length and the rate at which time passes both change appreciably as the object's speed approaches the speed of light, meaning that more variables and more complicated mathematics are required to calculate the object's motion. As a result, relativity requires the use of concepts such as vectors, tensors, pseudotensors and curvilinear coordinates.

In mathematics and physics, vector is a term that refers informally to some quantities that cannot be expressed by a single number, or to elements of some vector spaces.

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In geometry, an affine plane is a two-dimensional affine space.

References

  1. 1 2 3 "Details for IEV number 102-03-21: "vector quantity"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2024-09-07.
  2. ↑ "Details for IEV number 102-03-04: "vector"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2024-09-07.
  3. 1 2 Rao, A. (2006). Dynamics of Particles and Rigid Bodies: A Systematic Approach. Cambridge University Press. p. 3. ISBN   978-0-521-85811-3 . Retrieved 2024-09-08.
  4. 1 2 Teodorescu, Petre P. (2007-06-06). Mechanical Systems, Classical Models: Volume 1: Particle Mechanics. Springer Science & Business Media. ISBN   978-1-4020-5442-6.
  5. ↑ Merches, I.; Radu, D. (2014). Analytical Mechanics: Solutions to Problems in Classical Physics. CRC Press. p. 379. ISBN   978-1-4822-3940-9 . Retrieved 2024-09-09.
  6. ↑ Borisenko, A.I.; Tarapov, I.E.; Silverman, R.A. (2012). Vector and Tensor Analysis with Applications. Dover Books on Mathematics. Dover Publications. p. 2. ISBN   978-0-486-13190-0 . Retrieved 2024-09-08.
  7. ↑ "Appendix A. Linear Algebra from a Geometric Point of View". Differential Geometry: A Geometric Introduction. Ithaca, NY: David W. Henderson. 2013. p. 121–138. doi:10.3792/euclid/9781429799843-13. ISBN   978-1-4297-9984-3.
  8. 1 2 "ISO 80000-2:2019 - Quantities and units - Part 2: Mathematics". ISO. 2013-08-20. Retrieved 2024-09-08.
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