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| {{two other uses|the vectors mainly used in physics and engineering to represent directed quantities|mathematical vectors in general|Vector (mathematics and physics)||Vector (disambiguation){{!}}vector}}
| | They call me Emilia. One of the things he loves most is ice skating but he is struggling to find time for it. Years in the past we moved to North Dakota. For years he's been working as a receptionist.<br><br>Feel free to visit my page ... [http://www.allweb.com.kh/exalogwiki/?p=41948 over the counter std test] |
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| In [[mathematics]], [[physics]], and [[engineering]], a '''[[Euclidean]] vector''' (sometimes called a '''geometric'''<ref>{{harvnb|Ivanov|2001}}</ref> or '''spatial vector''',<ref>{{harvnb|Heinbockel|2001}}</ref> or—as here—simply a '''vector''') is a geometric quantity having [[Magnitude (mathematics)|magnitude]] (or [[euclidean norm|length]]) and [[Direction (geometry)|direction]] expressed numerically as [[tuple]]s [ x , y , z ] splitting the entire quantity into its orthogonal-axis components. A vector is an object that is an input for or an output from vector functions according to [[Linear algebra|vector algebra]]. A Euclidean vector is typically sketched as a directed [[line segment]], or arrow, connecting an ''initial point'' ''A'' with a ''terminal point'' ''B''<ref>{{harvnb|Ito|1993|p=1678}}; {{harvnb|Pedoe|1988}}</ref> and denoted by <math>\overrightarrow{AB}.</math> However, as an informational object, the vector is not as informative as a directed line segment (an ordered list of two points [ A , B ]) but rather expresses the displacement, or vector offset (change in location), A --> B. Technically, the [ x, y, z ] components of vector <math>\overrightarrow{AB}</math> are equal to the vector difference <math>\overrightarrow{B}</math> minus <math>\overrightarrow{A}</math>. In this way, the vector <math>\overrightarrow{AB}</math> considered as a numerical quantity conceals the locations of A and B while imparting the location of point B relative to A as if A were the coordinate origin.
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| Vectors play an important role in [[physics]]: [[velocity]] and [[acceleration]] of a moving object and [[force]]s acting on it are all described by vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, [[position (vector)|position]] or [[displacement (vector)|displacement]]), their magnitude and direction can be still represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the [[coordinate system]] used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include [[pseudovector]]s and [[tensor]]s.
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| It is important to distinguish Euclidean vectors from the more general concept in [[linear algebra]] of vectors as elements of a [[vector space]]. General vectors in this sense are fixed-size, ordered collections of items as in the case of Euclidean vectors, but the individual items may not be [[real number]]s, and the normal Euclidean concepts of length, distance and angle may not be applicable. (A vector space with a definition of these concepts is called an [[inner product space]].) In turn, both of these definitions of vector should be distinguished from the statistical concept of a [[random vector]]. The individual items in a random vector are individual real-valued [[random variable]]s, and are often manipulated using the same sort of mathematical vector and matrix operations that apply to the other types of vectors, but otherwise usually behave more like collections of individual values. Concepts of length, distance and angle do not normally apply to these vectors, either; rather, what links the values together is the potential [[correlation]]s among them.
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| The word "vector" originates from the Latin ''vehere'' meaning "to carry". It was first used by 18th century astronomers investigating planet rotation around the Sun.<ref>{{cite book|title=The Oxford english dictionary.|year=2001|publisher=Claredon Press|location=London|isbn=9780195219425|edition=2nd. ed.}}</ref>
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| ==History==
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| The concept of vector, as we know it today, evolved gradually over a period of more than 200 years. About a dozen people made significant contributions.<ref name="Crowe">Michael J. Crowe, [[A History of Vector Analysis]]; see also his [http://web.archive.org/web/20040126161844/http://www.nku.edu/~curtin/crowe_oresme.pdf lecture notes] on the subject.</ref> The immediate predecessor of vectors were [[quaternions]], devised by [[William Rowan Hamilton]] in 1843 as a generalization of complex numbers. Initially, his search was for a formalism to enable the analysis of three-dimensional space in the same way that [[complex number]]s had enabled analysis of two-dimensional space, but he arrived at a four-dimensional system. In 1846 Hamilton divided his quaternions into the sum of real and imaginary parts that he respectively called "scalar" and "vector":
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| :The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion.<ref>W. R. Hamilton (1846) ''London, Edinburgh & Dublin Philosophical Magazine'' 3rd series 29 27</ref>
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| Several other mathematicians developed vector-like systems around the same time as Hamilton including [[Giusto Bellavitis]], [[Augustin Cauchy]], [[Hermann Grassmann]], [[August Möbius]], [[Comte de Saint-Venant]], and [[Matthew O'Brien (mathematician)|Matthew O'Brien]]. Grassmann's 1840 work Theorie der Ebbe und Flut (Theory of the Ebb and Flow) was the first system of spatial analysis similar to today's system and had ideas corresponding to the cross product, scalar product and vector differentiation. Grassmann's work was largely neglected until the 1870s.<ref name="Crowe"/>
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| [[Peter Guthrie Tait]] carried the quaternion standard after Hamilton. His 1867 ''Elementary Treatise of Quaternions'' included extensive treatment of the nabla or [[del|del operator]] ∇.
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| In 1878 [[Elements of Dynamic]] was published by [[William Kingdon Clifford]]. Clifford simplified the quaternion study by isolating the [[dot product]] and [[cross product]] of two vectors from the complete quaternion product. This approach made vector calculations available to engineers and others working in three dimensions and skeptical of the fourth.
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| [[Josiah Willard Gibbs]], who was exposed to quaternions through [[James Clerk Maxwell]]'s ''Treatise on Electricity and Magnetism'', separated off their vector part for independent treatment. The first half of Gibbs's ''Elements of Vector Analysis'', published in 1881, presents what is essentially the modern system of vector analysis.<ref name="Crowe" /> In 1901 [[Edwin Bidwell Wilson]] published [[Vector Analysis]], adapted from Gibb's lectures, and banishing any mention of quaternions in the development of vector calculus.
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| ==Overview==
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| In [[physics]] and [[engineering]], a vector is typically regarded as a geometric entity characterized by a [[magnitude (mathematics)|magnitude]] and a direction. It is formally defined as a directed [[line segment]], or arrow, in a [[Euclidean space]].<ref>{{harvnb|Ito|1993|p=1678}}</ref> In [[pure mathematics]], a vector is defined more generally as any element of a [[vector space]]. In this context, vectors are abstract entities which may or may not be characterized by a magnitude and a direction. This generalized definition implies that the above mentioned geometric entities are a special kind of vectors, as they are elements of a special kind of vector space called [[Euclidean space]].
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| This article is about vectors strictly defined as arrows in Euclidean space. When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as '''geometric''', '''spatial''', or '''Euclidean''' vectors.
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| Being an arrow, a Euclidean vector possesses a definite ''initial point'' and ''terminal point''. A vector with fixed initial and terminal point is called a '''bound vector'''. When only the magnitude and direction of the vector matter, then the particular initial point is of no importance, and the vector is called a '''free vector'''. Thus two arrows <math>\overrightarrow{AB}</math> and <math>\overrightarrow{A'B'}</math> in space represent the same free vector if they have the same magnitude and direction: that is, they are equivalent if the quadrilateral ''ABB′A′'' is a [[parallelogram]]. If the Euclidean space is equipped with a choice of [[origin (mathematics)|origin]], then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin.
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| The term ''vector'' also has generalizations to higher dimensions and to more formal approaches with much wider applications.
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| ===Examples in one dimension===
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| Since the physicist's concept of [[force (physics)|force]] has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force ''F'' of 15 [[Newton (unit)|newtons]]. If the positive [[axis (mathematics)|axis]] is also directed rightward, then ''F'' is represented by the vector 15 N, and if positive points leftward, then the vector for ''F'' is −15 N. In either case, the magnitude of the vector is 15 N. Likewise, the vector representation of a displacement Δ''s'' of 4 [[meter (unit)|meters]] to the right would be 4 m or −4 m, and its magnitude would be 4 m regardless.
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| ===In physics and engineering===
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| Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to the rules of vector addition. An example is [[velocity]], the magnitude of which is [[speed]]. For example, the velocity ''5 meters per second upward'' could be represented by the vector (0,5) (in 2 dimensions with the positive ''y'' axis as 'up'). Another quantity represented by a vector is [[force]], since it has a magnitude and direction and follows the rules of vector addition. Vectors also describe many other physical quantities, such as linear displacement, [[displacement (vector)|displacement]], [[linear acceleration]], [[angular acceleration]], [[linear momentum]], and [[angular momentum]]. Other physical vectors, such as the [[electric field|electric]] and [[magnetic field]], are represented as a system of vectors at each point of a physical space; that is, a [[vector field]]. Examples of quantities that have magnitude and direction but fail to follow the rules of vector addition: Angular displacement and electric current. Consequently, these are not vectors.
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| ===In Cartesian space===
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| In the [[Cartesian coordinate system]], the simplest type of vector is a [[point vector]] (or position vector). It represents the displacement going from the origin ''O'' = (0,0,0) out to the point P = (x,y,z), and is equivalent numerically to point P's Cartesian coordinates (x,y,z). Point vectors are the starting point in vector geometry, i.e., other vector concepts assume point vectors as foundational objects.
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| What is the motion going in space from point A to point B? A 2-point vector is a convenient way to quantify spatial movement. This vector can be conceptualized by first assigning A as the initial point, and B as the terminal point. For instance, consider points ''A'' = (1,0,0) and ''B'' = (0,1,0). The vector <math>\overrightarrow{AB}</math> is drawn as an arrow connecting the point ''x''=1 on the ''x''-axis to the point ''y''=1 on the ''y''-axis. The meaning of vector <math>\overrightarrow{AB}</math> is the change in position, or motion, implied in moving from A --> B. The order of the two points is critical....moving from B --> A is a completely opposite movement, and thus a different vector quantity.
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| What numerics does vector <math>\overrightarrow{AB}</math> take on? Following the same example, the value of <math>\overrightarrow{AB}</math> is obtained by [[vector subtraction]]:
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| <math>\overrightarrow{AB}</math> ← <math>\overrightarrow{B}</math> - <math>\overrightarrow{A}</math>
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| (-1, 1, 0) ← (0,1,0) - (1,0,0)
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| This result can be interpreted to mean, starting at A, to get to point B go:
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| -1 along the x-axis
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| 1 along the y-axis
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| 0 along the z-axis
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| This motion vector was obtained numerically by doing vector subtraction <math>\overrightarrow{B}</math> - <math>\overrightarrow{A}</math>. It is sometimes useful to interpret vector <math>\overrightarrow{AB}</math> as location of B relative to A (the Cartesian coordinates B would take on if A were to become the new origin).
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| Using this logic, if you are a traveller and begin at location A, and your movement away from A is vector quantity <math>\overrightarrow{AB}</math> , where do you end up? You end up at location B, because <math>\overrightarrow{AB}</math> is the precise movement that takes you from A --> B. Numerically, a movement is modeled by [[vector addition]]:
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| <math>\overrightarrow{B}</math> ← <math>\overrightarrow{A}</math> + <math>\overrightarrow{AB}</math>
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| (0,1,0) ← (1,0,0) + (-1, 1, 0)
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| This same basic approach to representing motion can be extended to multi-point excursions through space. Consider this example: You are an airplane pilot, and take off from city A = (10, 20, 0), then fly to city B = (12, 15, 0), then fly to city C = (-3, -4, 0).
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| What motion vector do you fly to take you from C back to where you started at A? What motion gets you from C --> A? Call this quantity <math>\overrightarrow{CA}</math>, and calculate it taking the vector difference <math>\overrightarrow{A}</math> - <math>\overrightarrow{C}</math>. The motion you want is (13, 24, 0).
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| Now, suppose you take off from A, but only know the motion vectors you flew, not the absolute locations where you landed. You should be able to figure out where you are located nonetheless. The first flight segment was vector <math>\overrightarrow{AD}</math> = (16, -10, 0) landing at unknown location D. The second segment was vector <math>\overrightarrow{DE}</math> = (-8, 22, 0) landing at unknown location E. What is the location of city E? Successive motions can be lumped into a combined motion by vector addition. You want the Cartesian coordinates of E (its vector displacement from the origin), so you numerically combine all the motions that you would need to take starting at the origin to arrive to E by adding up the vector segments:
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| (10, 20, 0) = <math>\overrightarrow{A}</math> motion from origin to city A
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| + (16, -10, 0) = <math>\overrightarrow{AD}</math> motion from city A to city D
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| + (-8, 22, 0) = <math>\overrightarrow{DE}</math> motion from city D to city E
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| <nowiki>------------------------------------------------------------------------</nowiki>
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| (18, 32, 0) = <math>\overrightarrow{E}</math> motion from origin to city E
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| This style of aggregating successive known motions from a known starting point to estimate one's current spatial location is called [[dead reckoning]]. It was the technique used by the great ocean explorers to navigate between continents. An estimate of each day's travel was logged based on the day's heading and speed, and the motions were added together as demonstrated here.
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| Vectors may also be used to represent [[direction vector|directions]] in space, supplanting the use of slope and angles, to great advantage in 3D geometry. Their strength is the ability to obtain directions in space directly from pairs of Cartesian points, without resorting to angles and trigonometry.
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| Vector math is at the core of modern spatial software apps, including 3D animation, computer vision, robotics, GPS navigation, CAD, and protein modeling. The gradual ascendance of vector math representations over their scalar antecedents (e.g. slope, angle, trigonometric functions) is in part due to their ability to scale up naturally going from 2D --> 3D --> nD applications. The other major factor has been the advent of software-based computation since the 1960s, where vector representations impose fewer exceptions when writing algorithms compared to their scalar predecessors.
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| ===Euclidean and affine vectors===
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| In the geometrical and physical settings, sometimes it is possible to associate, in a natural way, a ''length'' or magnitude and a direction to vectors. In turn, the notion of direction is strictly associated with the notion of an ''angle'' between two vectors. When the length of vectors is defined, it is possible to also define a [[dot product]] — a scalar-valued product of two vectors — which gives a convenient algebraic characterization of both length (the square root of the dot product of a vector by itself) and angle (a function of the dot product between any two non-zero vectors). In three dimensions, it is further possible to define a [[cross product]] which supplies an algebraic characterization of the [[area]] and [[orientation (geometry)|orientation]] in space of the [[parallelogram]] defined by two vectors (used as sides of the parallelogram).
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| However, it is not always possible or desirable to define the length of a vector in a natural way. This more general type of spatial vector is the subject of [[vector space]]s (for bound vectors) and [[affine space]]s (for free vectors). An important example is [[Minkowski space]] that is important to our understanding of [[special relativity]], where there is a generalization of length that permits non-zero vectors to have zero length. Other physical examples come from [[thermodynamics]], where many of the quantities of interest can be considered vectors in a space with no notion of length or angle.<ref name="thermo-forms"
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| >[http://www.av8n.com/physics/thermo-forms.htm Thermodynamics and Differential Forms]</ref>
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| ===Generalizations===
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| In physics, as well as mathematics, a vector is often identified with a [[tuple]], or list of numbers, which depend on some auxiliary coordinate system or [[frame of reference|reference frame]]. When the coordinates are transformed, for example by rotation or stretching, then the components of the vector also transform. The vector itself has not changed, but the reference frame has, so the components of the vector (or measurements taken with respect to the reference frame) must change to compensate. The vector is called ''covariant'' or ''contravariant'' depending on how the transformation of the vector's components is related to the transformation of coordinates. In general, contravariant vectors are "regular vectors" with units of distance (such as a displacement) or distance times some other unit (such as velocity or acceleration); covariant vectors, on the other hand, have units of one-over-distance such as [[gradient]]. If you change units (a special case of a change of coordinates) from meters to milimeters, a scale factor of 1/1000, a displacement of 1 m becomes 1000 mm–a contravariant change in numerical value. In contrast, a gradient of 1 [[Kelvin|K]]/m becomes 0.001 K/mm–a covariant change in value. See [[covariance and contravariance of vectors]]. [[Tensor]]s are another type of quantity that behave in this way; in fact a vector is a special type of [[tensor]].
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| In pure [[mathematics]], a vector is any element of a [[vector space]] over some [[field (mathematics)|field]] and is often represented as a [[coordinate vector]]. The vectors described in this article are a very special case of this general definition because they are contravariant with respect to the ambient space. Contravariance captures the physical intuition behind the idea that a vector has "magnitude and direction".
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| ==Representations==
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| [[Image:vector from A to B.svg|right|200px|Vector arrow pointing from ''A'' to ''B'']]
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| Vectors are usually denoted in [[lowercase]] boldface, as '''a''' or lowercase italic boldface, as '''''a'''''. ([[Uppercase]] letters are typically used to represent [[matrix (mathematics)|matrices]].) Other conventions include <math>\vec{a}</math> or <u>''a''</u>, especially in handwriting. Alternatively, some use a [[tilde]] (~) or a wavy underline drawn beneath the symbol, e.g. <math>\underset{^\sim}a</math>, which is a convention for indicating boldface type. If the vector represents a directed [[distance]] or [[displacement (vector)|displacement]] from a point ''A'' to a point ''B'' (see figure), it can also be denoted as <math>\overrightarrow{AB}</math> or <u>''AB''</u>. Especially in literature in [[German language|German]] it was common to represent vectors with small [[fraktur]] letters as <math>\mathfrak{a}</math>.
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| Vectors are usually shown in graphs or other diagrams as arrows (directed [[line segment]]s), as illustrated in the figure. Here the point ''A'' is called the ''origin'', ''tail'', ''base'', or ''initial point''; point ''B'' is called the ''head'', ''tip'', ''endpoint'', ''terminal point'' or ''final point''. The length of the arrow is proportional to the vector's [[magnitude (mathematics)|magnitude]], while the direction in which the arrow points indicates the vector's direction.
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| [[Image:Notation for vectors in or out of a plane.svg|right|200px]]
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| On a two-dimensional diagram, sometimes a vector [[perpendicular]] to the [[plane (mathematics)|plane]] of the diagram is desired. These vectors are commonly shown as small circles. A circle with a dot at its centre (Unicode U+2299 ⊙) indicates a vector pointing out of the front of the diagram, toward the viewer. A circle with a cross inscribed in it (Unicode U+2297 ⊗) indicates a vector pointing into and behind the diagram. These can be thought of as viewing the tip of an [[arrow (weapon)|arrow]] head on and viewing the flights of an arrow from the back.
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| [[Image:Position vector.svg|thumb|right|A vector in the Cartesian plane, showing the position of a point ''A'' with coordinates (2,3).]]
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| [[Image:3D Vector.svg|300px|right]]
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| In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an ''n''-dimensional Euclidean space can be represented as [[coordinate vector]]s in a [[Cartesian coordinate system]]. The endpoint of a vector can be identified with an ordered list of ''n'' real numbers (''n''-[[tuple]]). These numbers are the [[Cartesian coordinate|coordinates]] of the endpoint of the vector, with respect to a given [[Cartesian coordinate system]], and are typically called the '''[[scalar component]]s''' (or '''scalar projections''') of the vector on the axes of the coordinate system.
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| As an example in two dimensions (see figure), the vector from the origin ''O'' = (0,0) to the point ''A'' = (2,3) is simply written as
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| :<math>\mathbf{a} = (2,3).</math>
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| The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation <math>\overrightarrow{OA}</math> is usually not deemed necessary and very rarely used.
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| In ''three dimensional'' Euclidean space (or {{math|'''R'''<sup>''3''</sup>}}), vectors are identified with triples of scalar components:
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| :<math>\mathbf{a} = (a_1, a_2, a_3).</math>
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| :also written
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| :<math>\mathbf{a} = (a_\text{x}, a_\text{y}, a_\text{z}).</math>
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| This can be generalised to ''n-dimensional'' Euclidean space (or {{math|'''R'''<sup>''n''</sup>}}).
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| :<math>\mathbf{a} = (a_1, a_2, a_3, \cdots, a_{n-1}, a_n).</math>
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| These numbers are often arranged into a [[column vector]] or [[row vector]], particularly when dealing with [[matrix (mathematics)|matrices]], as follows:
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| :<math>\mathbf{a} = \begin{bmatrix}
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| a_1\\
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| a_2\\
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| a_3\\
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| \end{bmatrix}
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| </math>
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| :<math>\mathbf{a} = [ a_1\ a_2\ a_3 ].</math>
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| Another way to represent a vector in ''n''-dimensions is to introduce the [[standard basis]] vectors. For instance, in three dimensions, there are three of them:
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| :<math>{\mathbf e}_1 = (1,0,0),\ {\mathbf e}_2 = (0,1,0),\ {\mathbf e}_3 = (0,0,1).</math>
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| These have the intuitive interpretation as vectors of unit length pointing up the ''x'', ''y'', and ''z'' axis of a [[Cartesian coordinate system]], respectively. In terms of these, any vector '''a''' in {{math|'''R'''<sup>''3''</sup>}} can be expressed in the form:
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| :<math>\mathbf{a} = (a_1,a_2,a_3) = a_1(1,0,0) + a_2(0,1,0) + a_3(0,0,1), \ </math>
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| or
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| :<math>\mathbf{a} = \mathbf{a}_1 + \mathbf{a}_2 + \mathbf{a}_3 = a_1{\mathbf e}_1 + a_2{\mathbf e}_2 + a_3{\mathbf e}_3,</math>
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| where '''a'''<sub>1</sub>, '''a'''<sub>2</sub>, '''a'''<sub>3</sub> are called the '''[[vector component]]s''' (or '''vector projections''') of '''a''' on the basis vectors or, equivalently, on the corresponding Cartesian axes ''x'', ''y'', and ''z'' (see figure), while ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub> are the respective [[scalar component]]s (or scalar projections).
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| In introductory physics textbooks, the standard basis vectors are often instead denoted <math>\mathbf{i},\mathbf{j},\mathbf{k}</math> (or <math>\mathbf{\hat{x}}, \mathbf{\hat{y}}, \mathbf{\hat{z}}</math>, in which the [[hat symbol]] '''^''' typically denotes [[unit vector]]s). In this case, the scalar and vector components are denoted respectively ''a''<sub>x</sub>, ''a''<sub>y</sub>, ''a''<sub>z</sub>, and '''a'''<sub>x</sub>, '''a'''<sub>y</sub>, '''a'''<sub>z</sub> (note the difference in boldface). Thus,
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| :<math>\mathbf{a} = \mathbf{a}_\text{x} + \mathbf{a}_\text{y} + \mathbf{a}_\text{z} = a_\text{x}{\mathbf i} + a_\text{y}{\mathbf j} + a_\text{z}{\mathbf k}.</math>
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| The notation '''e'''<sub>''i''</sub> is compatible with the [[index notation]] and the [[summation convention]] commonly used in higher level mathematics, physics, and engineering.
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| ===Decomposition===
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| {{details|Basis (linear algebra)}}
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| As explained [[Euclidean vector#Representations|above]] a vector is often described by a set of vector components that [[#Addition and subtraction|add up]] to form the given vector. Typically, these components are the [[Vector projection|projections]] of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be ''decomposed'' or ''resolved with respect to'' that set.
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| [[Image:Normal and tangent illustration.png|right|thumb|Illustration of tangential and normal components of a vector to a surface.]]
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| However, the decomposition of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected.
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| Moreover, the use of Cartesian unit vectors such as <math>\mathbf{\hat{x}}, \mathbf{\hat{y}}, \mathbf{\hat{z}}</math> as a [[Basis (linear algebra)|basis]] in which to represent a vector is not mandated. Vectors can also be expressed in terms of the unit vectors of a [[cylindrical coordinate system]] (<math>\boldsymbol{\hat{\rho}}, \boldsymbol{\hat{\phi}}, \mathbf{\hat{z}}</math>) or [[spherical coordinate system]] (<math>\mathbf{\hat{r}}, \boldsymbol{\hat{\theta}}, \boldsymbol{\hat{\phi}}</math>). The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry respectively.
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| The choice of a coordinate system doesn't affect the properties of a vector or its behaviour under transformations.
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| A vector can be also decomposed with respect to "non-fixed" axes which change their [[orientation (geometry)|orientation]] as a function of time or space. For example, a vector in three-dimensional space can be decomposed with respect to two axes, respectively ''normal'', and ''tangent'' to a surface (see figure).
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| Moreover, the ''radial'' and ''[[tangential component]]s'' of a vector relate to the ''[[radius]] of [[rotation]]'' of an object. The former is [[Parallel (geometry)|parallel]] to the radius and the latter is [[Perpendicular|orthogonal]] to it.<ref>[http://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.angacc.html U. Guelph Physics Dept., "Torque and Angular Acceleration"]</ref>
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| In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., a ''global'' coordinate system, or [[inertial reference frame]]).
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| ==Basic properties==
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| The following section uses the [[Cartesian coordinate system]] with basis vectors
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| :<math>{\mathbf e}_1 = (1,0,0),\ {\mathbf e}_2 = (0,1,0),\ {\mathbf e}_3 = (0,0,1)</math>
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| and assumes that all vectors have the origin as a common base point. A vector '''a''' will be written as
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| :<math>{\mathbf a} = a_1{\mathbf e}_1 + a_2{\mathbf e}_2 + a_3{\mathbf e}_3.</math>
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| ===Equality===
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| Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors
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| :<math>{\mathbf a} = a_1{\mathbf e}_1 + a_2{\mathbf e}_2 + a_3{\mathbf e}_3</math>
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| and
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| :<math>{\mathbf b} = b_1{\mathbf e}_1 + b_2{\mathbf e}_2 + b_3{\mathbf e}_3</math>
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| are equal if
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| :<math>a_1 = b_1,\quad a_2=b_2,\quad a_3=b_3.\,</math>
| |
| | |
| ===Addition and subtraction===
| |
| {{details|Vector space}}
| |
| Assume now that '''a''' and '''b''' are not necessarily equal vectors, but that they may have different magnitudes and directions. The sum of '''a''' and '''b''' is
| |
| :<math>\mathbf{a}+\mathbf{b}
| |
| =(a_1+b_1)\mathbf{e}_1
| |
| +(a_2+b_2)\mathbf{e}_2
| |
| +(a_3+b_3)\mathbf{e}_3.</math>
| |
| | |
| The addition may be represented graphically by placing the tail of the arrow '''b''' at the head of the arrow '''a''', and then drawing an arrow from the tail of '''a''' to the head of '''b'''. The new arrow drawn represents the vector '''a''' + '''b''', as illustrated below:
| |
| | |
| [[Image:Vector addition.svg|250px|center|The addition of two vectors '''a''' and '''b''']]
| |
| | |
| This addition method is sometimes called the ''parallelogram rule'' because '''a''' and '''b''' form the sides of a [[parallelogram]] and '''a''' + '''b''' is one of the diagonals. If '''a''' and '''b''' are bound vectors that have the same base point, this point will also be the base point of '''a''' + '''b'''. One can check geometrically that '''a''' + '''b''' = '''b''' + '''a''' and ('''a''' + '''b''') + '''c''' = '''a''' + ('''b''' + '''c''').
| |
| | |
| The difference of '''a''' and '''b''' is
| |
| | |
| :<math>\mathbf{a}-\mathbf{b}
| |
| =(a_1-b_1)\mathbf{e}_1
| |
| +(a_2-b_2)\mathbf{e}_2
| |
| +(a_3-b_3)\mathbf{e}_3.</math>
| |
| | |
| Subtraction of two vectors can be geometrically defined as follows: to subtract '''b''' from '''a''', place the tails of '''a''' and '''b''' at the same point, and then draw an arrow from the head of '''b''' to the head of '''a'''. This new arrow represents the vector '''a''' − '''b''', as illustrated below:
| |
| | |
| [[Image:Vector subtraction.svg|125px|center|The subtraction of two vectors '''a''' and '''b''']]
| |
| | |
| Subtraction of two vectors may also be performed by adding the opposite of the second vector to the first vector, that is, '''a''' − '''b''' = '''a''' + (−'''b''').
| |
| | |
| ===Scalar multiplication===
| |
| {{main|Scalar multiplication}}
| |
| | |
| [[Image:Scalar multiplication by r=3.svg|250px|thumb|right|Scalar multiplication of a vector by a factor of 3 stretches the vector out.]]
| |
| [[Image:Scalar multiplication of vectors2.svg|250px|thumb|right|The scalar multiplications −'''a''' and 2'''a''' of a vector '''a''']]
| |
| A vector may also be multiplied, or re-''scaled'', by a [[real number]] ''r''. In the context of [[vector analysis|conventional vector algebra]], these real numbers are often called '''scalars''' (from ''scale'') to distinguish them from vectors. The operation of multiplying a vector by a scalar is called ''scalar multiplication''. The resulting vector is
| |
| | |
| :<math>r\mathbf{a}=(ra_1)\mathbf{e}_1
| |
| +(ra_2)\mathbf{e}_2
| |
| +(ra_3)\mathbf{e}_3.</math>
| |
| | |
| Intuitively, multiplying by a scalar ''r'' stretches a vector out by a factor of ''r''. Geometrically, this can be visualized (at least in the case when ''r'' is an integer) as placing ''r'' copies of the vector in a line where the endpoint of one vector is the initial point of the next vector.
| |
| | |
| If ''r'' is negative, then the vector changes direction: it flips around by an angle of 180°. Two examples (''r'' = −1 and ''r'' = 2) are given below:
| |
| | |
| Scalar multiplication is [[Distributivity|distributive]] over vector addition in the following sense: ''r''('''a''' + '''b''') = ''r'''''a''' + ''r'''''b''' for all vectors '''a''' and '''b''' and all scalars ''r''. One can also show that '''a''' − '''b''' = '''a''' + (−1)'''b'''.
| |
| <!--
| |
| The set of all geometrical vectors, together with the operations of vector addition and scalar multiplication, satisfies all the axioms of a [[vector space]]. Similarly, the set of all bound vectors with a common base point forms a vector space. This is where the term "vector space" originated.
| |
| -->
| |
| | |
| ===Length===<!-- This section is linked from [[Law of cosines]] -->
| |
| The ''[[length]]'' or ''[[Magnitude (mathematics)|magnitude]]'' or ''[[Norm (mathematics)|norm]]'' of the vector '''a''' is denoted by ‖'''a'''‖ or, less commonly, |'''a'''|, which is not to be confused with the [[absolute value]] (a scalar "norm").
| |
| | |
| The length of the vector '''a''' can be computed with the [[Norm (mathematics)#Euclidean norm|Euclidean norm]]
| |
| | |
| :<math>\left\|\mathbf{a}\right\|=\sqrt{{a_1}^2+{a_2}^2+{a_3}^2}</math>
| |
| | |
| which is a consequence of the [[Pythagorean theorem]] since the basis vectors '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> are orthogonal unit vectors.
| |
| | |
| This happens to be equal to the square root of the [[dot product]], discussed below, of the vector with itself:
| |
| | |
| :<math>\left\|\mathbf{a}\right\|=\sqrt{\mathbf{a}\cdot\mathbf{a}}.</math>
| |
| | |
| ;Unit vector
| |
| [[Image:Vector normalization.svg|thumb|right|The normalization of a vector '''a''' into a unit vector '''â''']]
| |
| {{main|Unit vector}}
| |
| | |
| A ''unit vector'' is any vector with a length of one; normally unit vectors are used simply to indicate direction. A vector of arbitrary length can be divided by its length to create a unit vector. This is known as ''normalizing'' a vector. A unit vector is often indicated with a hat as in '''â'''.
| |
| | |
| To normalize a vector '''a''' = [''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>], scale the vector by the reciprocal of its length ||'''a'''||. That is:
| |
| | |
| :<math>\mathbf{\hat{a}} = \frac{\mathbf{a}}{\left\|\mathbf{a}\right\|} = \frac{a_1}{\left\|\mathbf{a}\right\|}\mathbf{e}_1 + \frac{a_2}{\left\|\mathbf{a}\right\|}\mathbf{e}_2 + \frac{a_3}{\left\|\mathbf{a}\right\|}\mathbf{e}_3</math>
| |
| | |
| ;Null vector
| |
| {{main|Null vector}}
| |
| | |
| The ''null vector'' (or ''zero vector'') is the vector with length zero. Written out in coordinates, the vector is (0,0,0), and it is commonly denoted <math>\vec{0}</math>, or '''0''', or simply 0. Unlike any other vector it has an arbitrary or indeterminate direction, and cannot be normalized (that is, there is no unit vector which is a multiple of the null vector). The sum of the null vector with any vector '''a''' is '''a''' (that is, '''0'''+'''a'''='''a''').
| |
| | |
| ===Dot product===
| |
| {{main|dot product}}
| |
| | |
| The ''dot product'' of two vectors '''a''' and '''b''' (sometimes called the ''[[inner product space|inner product]]'', or, since its result is a scalar, the ''scalar product'') is denoted by '''a''' ∙ '''b''' and is defined as:
| |
| | |
| :<math>\mathbf{a}\cdot\mathbf{b}
| |
| =\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta</math>
| |
| | |
| where ''θ'' is the measure of the [[angle]] between '''a''' and '''b''' (see [[trigonometric function]] for an explanation of cosine). Geometrically, this means that '''a''' and '''b''' are drawn with a common start point and then the length of '''a''' is multiplied with the length of that component of '''b''' that points in the same direction as '''a'''.
| |
| | |
| The dot product can also be defined as the sum of the products of the components of each vector as
| |
| | |
| :<math>\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3.</math>
| |
| | |
| ===Cross product===
| |
| {{main|Cross product}}
| |
| | |
| The ''cross product'' (also called the ''vector product'' or ''outer product'') is only meaningful in three or [[Seven-dimensional cross product|seven]] dimensions. The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector. The cross product, denoted '''a''' × '''b''', is a vector perpendicular to both '''a''' and '''b''' and is defined as
| |
| | |
| :<math>\mathbf{a}\times\mathbf{b}
| |
| =\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\sin(\theta)\,\mathbf{n}</math>
| |
| | |
| where ''θ'' is the measure of the angle between '''a''' and '''b''', and '''n''' is a unit vector [[perpendicular]] to both '''a''' and '''b''' which completes a [[Right hand rule|right-handed]] system. The right-handedness constraint is necessary because there exist ''two'' unit vectors that are perpendicular to both '''a''' and '''b''', namely, '''n''' and (–'''n''').
| |
| [[Image:Cross product vector.svg|thumb|right|An illustration of the cross product]]
| |
| | |
| The cross product '''a''' × '''b''' is defined so that '''a''', '''b''', and '''a''' × '''b''' also becomes a right-handed system (but note that '''a''' and '''b''' are not necessarily [[orthogonal]]). This is the [[right-hand rule]].
| |
| | |
| The length of '''a''' × '''b''' can be interpreted as the area of the parallelogram having '''a''' and '''b''' as sides.
| |
| | |
| The cross product can be written as
| |
| :<math>{\mathbf a}\times{\mathbf b} = (a_2 b_3 - a_3 b_2) {\mathbf e}_1 + (a_3 b_1 - a_1 b_3) {\mathbf e}_2 + (a_1 b_2 - a_2 b_1) {\mathbf e}_3.</math>
| |
| | |
| For arbitrary choices of spatial orientation (that is, allowing for left-handed as well as right-handed coordinate systems) the cross product of two vectors is a [[pseudovector]] instead of a vector (see below).
| |
| | |
| ===Scalar triple product===
| |
| {{main|Triple product#Scalar triple product|l1=Scalar triple product}}
| |
| The ''scalar triple product'' (also called the ''box product'' or ''mixed triple product'') is not really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is sometimes denoted by ('''a''' '''b''' '''c''') and defined as:
| |
| | |
| :<math>(\mathbf{a}\ \mathbf{b}\ \mathbf{c})
| |
| =\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}).</math>
| |
| | |
| It has three primary uses. First, the absolute value of the box product is the volume of the [[parallelepiped]] which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are [[linear independence|linearly dependent]], which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors '''a''', '''b''' and '''c''' are right-handed.
| |
| | |
| In components (''with respect to a right-handed orthonormal basis''), if the three vectors are thought of as rows (or columns, but in the same order), the scalar triple product is simply the [[determinant]] of the 3-by-3 [[Matrix (mathematics)|matrix]] having the three vectors as rows
| |
| :<math>(\mathbf{a}\ \mathbf{b}\ \mathbf{c})=\left|\begin{pmatrix}
| |
| a_1 & a_2 & a_3 \\
| |
| b_1 & b_2 & b_3 \\
| |
| c_1 & c_2 & c_3 \\
| |
| \end{pmatrix}\right|</math>
| |
| | |
| The scalar triple product is linear in all three entries and anti-symmetric in the following sense:
| |
| :<math>
| |
| (\mathbf{a}\ \mathbf{b}\ \mathbf{c}) = (\mathbf{c}\ \mathbf{a}\ \mathbf{b}) = (\mathbf{b}\ \mathbf{c}\ \mathbf{a})=
| |
| -(\mathbf{a}\ \mathbf{c}\ \mathbf{b}) = -(\mathbf{b}\ \mathbf{a}\ \mathbf{c}) = -(\mathbf{c}\ \mathbf{b}\ \mathbf{a}).</math>
| |
| | |
| ===Multiple Cartesian bases===
| |
| All examples thus far have dealt with vectors expressed in terms of the same basis, namely, '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>. However, a 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 vector. For example, using the vector '''a''' from above,
| |
| | |
| :<math>
| |
| \mathbf{a} =
| |
| a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3 =
| |
| u\mathbf{n}_1 + v\mathbf{n}_2 + w\mathbf{n}_3
| |
| </math>
| |
| | |
| where '''n'''<sub>1</sub>, '''n'''<sub>2</sub>, '''n'''<sub>3</sub> form another orthonormal basis not aligned with '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>. The values of ''u'', ''v'', and ''w'' are such that the resulting vector sum is exactly '''a'''.
| |
| | |
| It is not uncommon to encounter vectors known in terms of different bases (for example, one basis fixed to the Earth and a second basis fixed to a moving vehicle). In order to perform many of the operations defined above, it is necessary to know the vectors in terms of the same basis. One simple way to express a vector known in one basis in terms of another uses column matrices that represent the vector in each basis along with a third matrix containing the information that relates the two bases. For example, in order to find the values of ''u'', ''v'', and ''w'' that define '''a''' in the '''n'''<sub>1</sub>, '''n'''<sub>2</sub>, '''n'''<sub>3</sub> basis, a matrix multiplication may be employed in the form
| |
| | |
| :<math>\begin{bmatrix} u \\ v \\ w \\ \end{bmatrix} = \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} </math>
| |
| | |
| where each matrix element ''c''<sub>''jk''</sub> is the [[Direction cosine#Cartesian coordinates|direction cosine]] relating '''n'''<sub>''j''</sub> to '''e'''<sub>''k''</sub>.<ref name="dynon16">{{harvnb|Kane|Levinson|1996|pp=20–22}}</ref> The term ''direction cosine'' refers to the [[cosine]] of the angle between two unit vectors, which is also equal to their [[#Dot product|dot product]].<ref name="dynon16"/>
| |
| | |
| By referring collectively to '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> as the ''e'' basis and to '''n'''<sub>1</sub>, '''n'''<sub>2</sub>, '''n'''<sub>3</sub> as the ''n'' basis, the matrix containing all the ''c''<sub>''jk''</sub> is known as the "'''[[transformation matrix]]''' from ''e'' to ''n''", or the "'''[[rotation matrix]]''' from ''e'' to ''n''" (because it can be imagined as the "rotation" of a vector from one basis to another), or the "'''direction cosine matrix''' from ''e'' to ''n''"<ref name="dynon16"/> (because it contains direction cosines).
| |
| | |
| The properties of a [[rotation matrix]] are such that its [[matrix inverse|inverse]] is equal to its [[matrix transpose|transpose]]. This means that the "rotation matrix from ''e'' to ''n''" is the transpose of "rotation matrix from ''n'' to ''e''".
| |
| | |
| By applying several matrix multiplications in succession, any vector can be expressed in any basis so long as the set of direction cosines is known relating the successive bases.<ref name="dynon16"/>
| |
| | |
| ===Other dimensions===
| |
| With the exception of the cross and triple products, the above formulae generalise to two dimensions and higher dimensions. For example, addition generalises to two dimensions as
| |
| :<math>(a_1{\mathbf e}_1 + a_2{\mathbf e}_2)+(b_1{\mathbf e}_1 + b_2{\mathbf e}_2) = (a_1+b_1){\mathbf e}_1 + (a_2+b_2){\mathbf e}_2</math>
| |
| and in four dimensions as
| |
| :<math>\begin{align}(a_1{\mathbf e}_1 + a_2{\mathbf e}_2 + a_3{\mathbf e}_3 + a_4{\mathbf e}_4) &+ (b_1{\mathbf e}_1 + b_2{\mathbf e}_2 + b_3{\mathbf e}_3 + b_4{\mathbf e}_4) =\\
| |
| (a_1+b_1){\mathbf e}_1 + (a_2+b_2){\mathbf e}_2 &+ (a_3+b_3){\mathbf e}_3 + (a_4+b_4){\mathbf e}_4.\end{align}</math>
| |
| | |
| The cross product does not readily generalise to other dimensions, though the closely related [[Exterior algebra#Areas in the plane|exterior product]] does, whose result is a [[bivector]]. In two dimensions this is simply a [[pseudoscalar]]
| |
| :<math>(a_1{\mathbf e}_1 + a_2{\mathbf e}_2)\wedge(b_1{\mathbf e}_1 + b_2{\mathbf e}_2) = (a_1 b_2 - a_2 b_1)\mathbf{e}_1 \mathbf{e}_2.</math>
| |
| | |
| A [[seven-dimensional cross product]] is similar to the cross product in that its result is a vector orthogonal to the two arguments; there is however no natural way of selecting one of the possible such products.
| |
| | |
| ==Physics==
| |
| Vectors have many uses in physics and other sciences.
| |
| | |
| ===Length and units===
| |
| In abstract vector spaces, the length of the arrow depends on a [[Dimensionless number|dimensionless]] [[Scale (measurement)|scale]]. If it represents, for example, a force, the "scale" is of [[Dimensional analysis|physical dimension]] length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2 cm, the scales are 1:250 and 1 m:50 N respectively. Equal length of vectors of different dimension has no particular significance unless there is some [[proportionality constant]] inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance.
| |
| | |
| ===Vector-valued functions===
| |
| {{main|Vector-valued function}}
| |
| Often in areas of physics and mathematics, a vector evolves in time, meaning that it depends on a time parameter ''t''. For instance, if '''r''' represents the position vector of a particle, then '''r'''(''t'') gives a [[parametric equation|parametric]] representation of the trajectory of the particle. Vector-valued functions can be [[derivative|differentiated]] and [[integral|integrated]] by differentiating or integrating the components of the vector, and many of the familiar rules from [[calculus]] continue to hold for the derivative and integral of vector-valued functions.
| |
| | |
| ===Position, velocity and acceleration===
| |
| The position of a point '''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) in three-dimensional space can be represented as a [[position vector]] whose base point is the origin
| |
| :<math>{\mathbf x} = x_1 {\mathbf e}_1 + x_2{\mathbf e}_2 + x_3{\mathbf e}_3.</math>
| |
| The position vector has dimensions of [[length]].
| |
| | |
| Given two points '''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>), '''y''' = (''y''<sub>1</sub>, ''y''<sub>2</sub>, ''y''<sub>3</sub>) their [[Displacement (vector)|displacement]] is a vector
| |
| :<math>{\mathbf y}-{\mathbf x}=(y_1-x_1){\mathbf e}_1 + (y_2-x_2){\mathbf e}_2 + (y_3-x_3){\mathbf e}_3.</math>
| |
| which specifies the position of ''y'' relative to ''x''. The length of this vector gives the straight line distance from ''x'' to ''y''. Displacement has the dimensions of length.
| |
| | |
| The [[velocity]] '''v''' of a point or particle is a vector, its length gives the [[speed]]. For constant velocity the position at time ''t'' will be
| |
| :<math>{\mathbf x}_t= t {\mathbf v} + {\mathbf x}_0,</math>
| |
| where '''x'''<sub>0</sub> is the position at time ''t''=0. Velocity is the [[Euclidean vector#Ordinary derivative|time derivative]] of position. Its dimensions are length/time.
| |
| | |
| [[Acceleration]] '''a''' of a point is vector which is the [[Euclidean vector#Ordinary derivative|time derivative]] of velocity. Its dimensions are length/time<sup>2</sup>.
| |
| | |
| ===Force, energy, work===
| |
| [[Force]] is a vector with dimensions of mass×length/time<sup>2</sup> and [[Newton's laws of motion#Newton's second law|Newton's second law]] is the scalar multiplication
| |
| :<math>{\mathbf F} = m{\mathbf a}</math>
| |
| | |
| Work is the dot product of [[force]] and [[displacement (vector)|displacement]]
| |
| :<math>E = {\mathbf F} \cdot ({\mathbf x}_2 - {\mathbf x}_1).</math>
| |
| <!--
| |
| In physics, scalars may also have a unit of measurement associated with them. For instance, [[Newton's laws of motion#Newton's second law|Newton's second law]] is
| |
| :<math>{\mathbf F} = m{\mathbf a}</math>
| |
| where '''F''' has units of force, '''a''' has units of acceleration, and the scalar ''m'' has units of mass. In one possible physical interpretation of the above diagram, the scale of acceleration is, for instance, 2 m/s<sup>2</sup> : cm, and that of force 5 N : cm. Thus a scale ratio of 2.5 kg : 1 is used for mass. Similarly, if displacement has a scale of 1:1000 and velocity of 0.2 cm : 1 m/s, or equivalently, 2 ms : 1, a scale ratio of 0.5 : s is used for time.
| |
| -->
| |
| | |
| ==Vectors as directional derivatives==
| |
| A vector may also be defined as a ''[[directional derivative]]'': consider a [[function (mathematics)|function]] <math>f(x^\alpha)</math> and a curve <math>x^\alpha (\tau)</math>. Then the directional derivative of <math>f</math> is a scalar defined as
| |
| | |
| :<math>\frac{df}{d\tau} = \sum_{\alpha=1}^n \frac{dx^\alpha}{d\tau}\frac{\partial f}{\partial x^\alpha}.</math>
| |
| | |
| where the index <math>\alpha</math> is [[Summation convention|summed over]] the appropriate number of dimensions (for example, from 1 to 3 in 3-dimensional Euclidean space, from 0 to 3 in 4-dimensional spacetime, etc.). Then consider a vector tangent to <math>x^\alpha (\tau)</math>:
| |
| | |
| :<math>t^\alpha = \frac{dx^\alpha}{d\tau}.</math>
| |
| | |
| The directional derivative can be rewritten in differential form (without a given function <math>f</math>) as
| |
| | |
| :<math>\frac{d}{d\tau} = \sum_\alpha t^\alpha\frac{\partial}{\partial x^\alpha}.</math>
| |
| | |
| Therefore any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative. A vector can therefore be defined precisely as
| |
| | |
| :<math>\mathbf{a} \equiv a^\alpha \frac{\partial}{\partial x^\alpha}.</math>
| |
| | |
| ==Vectors, pseudovectors, and transformations==
| |
| An alternative characterization of Euclidean vectors, especially in physics, describes them as lists of quantities which behave in a certain way under a [[coordinate system|coordinate transformation]]. A ''contravariant vector'' is required to have components that "transform like the coordinates" under changes of coordinates such as [[rotation (mathematics)|rotation]] and dilation. The vector itself does not change under these operations; instead, the components of the vector make a change that cancels the change in the spatial axes, in the same way that co-ordinates change. In other words, if the reference axes were rotated in one direction, the component representation of the vector would rotate in exactly the opposite way. Similarly, if the reference axes were stretched in one direction, the components of the vector, like the co-ordinates, would reduce in an exactly compensating way. Mathematically, if the coordinate system undergoes a transformation described by an [[invertible matrix]] ''M'', so that a coordinate vector '''x''' is transformed to '''x'''′ = ''M'''''x''', then a contravariant vector '''v''' must be similarly transformed via '''v'''′ = ''M'''''v'''. This important requirement is what distinguishes a contravariant vector from any other triple of physically meaningful quantities. For example, if ''v'' consists of the ''x'', ''y'', and ''z''-components of [[velocity]], then ''v'' is a contravariant vector: if the coordinates of space are stretched, rotated, or twisted, then the components of the velocity transform in the same way. On the other hand, for instance, a triple consisting of the length, width, and height of a rectangular box could make up the three components of an abstract [[vector space|vector]], but this vector would not be contravariant, since rotating the box does not change the box's length, width, and height. Examples of contravariant vectors include [[displacement (vector)|displacement]], [[velocity]], [[electric field]], [[momentum]], [[force]], and [[acceleration]].
| |
| | |
| In the language of [[differential geometry]], the requirement that the components of a vector transform according to the same matrix of the coordinate transition is equivalent to defining a ''contravariant vector'' to be a [[tensor]] of [[Covariance and contravariance of vectors|contravariant]] rank one. Alternatively, a contravariant vector is defined to be a [[tangent space|tangent vector]], and the rules for transforming a contravariant vector follow from the [[chain rule]].
| |
| | |
| Some vectors transform like contravariant vectors, except that when they are reflected through a mirror, they flip ''and'' gain a minus sign. A transformation that switches right-handedness to left-handedness and vice versa like a mirror does is said to change the ''[[orientation (mathematics)|orientation]]'' of space. A vector which gains a minus sign when the orientation of space changes is called a ''[[pseudovector]]'' or an ''axial vector''. Ordinary vectors are sometimes called ''true vectors'' or ''polar vectors'' to distinguish them from pseudovectors. Pseudovectors occur most frequently as the [[cross product]] of two ordinary vectors.
| |
| | |
| One example of a pseudovector is [[angular velocity]]. Driving in a [[car]], and looking forward, each of the [[wheel]]s has an angular velocity vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the ''reflection'' of this angular velocity vector points to the right, but the ''actual'' angular velocity vector of the wheel still points to the left, corresponding to the minus sign. Other examples of pseudovectors include [[magnetic field]], [[torque]], or more generally any cross product of two (true) vectors.
| |
| | |
| This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying [[symmetry]] properties. See [[parity (physics)]].
| |
| | |
| ==See also==
| |
| * [[Affine space]], which distinguishes between vectors and [[Point (geometry)|points]]
| |
| * [[Array data structure]] or [[Vector (Computer Science)]]
| |
| * [[Banach space]]
| |
| * [[Clifford algebra]]
| |
| * [[Complex number]]
| |
| * [[Coordinate system]]
| |
| * [[Covariance and contravariance of vectors]]
| |
| * [[Four-vector]], a non-Euclidean vector in Minkowski space (i.e. four-dimensional spacetime), important in [[theory of relativity|relativity]]
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| * [[Function space]]
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| * [[Grassmann|Grassmann's]] ''Ausdehnungslehre''
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| * [[Hilbert space]]
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| * [[Normal vector]]
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| * [[Null vector]]
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| * [[Pseudovector]]
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| * [[Quaternion]]
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| * [[Tangential and normal components]] (of a vector)
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| * [[Tensor]]
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| * [[Unit vector]]
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| * [[Vector bundle]]
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| * [[Vector calculus]]
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| * [[Vector notation]]
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| * [[Vector-valued function]]
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| | |
| ==Notes==
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| {{reflist|2}}
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| | |
| ==References==
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| '''Mathematical treatments'''
| |
| * {{cite book|author = [[Tom Apostol|Apostol, T.]]|title=Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra|publisher=John Wiley and Sons|year=1967|isbn= 978-0-471-00005-1}}
| |
| * {{cite book|author = Apostol, T.|title=Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications|publisher = John Wiley and Sons|year=1969|isbn=978-0-471-00007-5}}
| |
| * {{citation|last1=Kane|first1=Thomas R.|last2=Levinson|first2=David A.|title=Dynamics Online|publisher=OnLine Dynamics, Inc.|location=Sunnyvale, California|year=1996}}
| |
| * {{citation|title=Introduction to Tensor Calculus and Continuum Mechanics|first=J. H.|last=Heinbockel|publisher=Trafford Publishing|year=2001|isbn=1-55369-133-4|url=http://www.math.odu.edu/~jhh/counter2.html}}
| |
| * {{Citation | last1=Ito | first1=Kiyosi | title=Encyclopedic Dictionary of Mathematics | publisher=[[MIT Press]] | edition=2nd | isbn=978-0-262-59020-4 | year=1993}}
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| * {{springer|id=V/v096340|title=Vector, geometric|first=A.B.|last=Ivanov}}
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| * {{cite book|author = [[Daniel Pedoe|Pedoe, D.]]|title = Geometry: A comprehensive course|publisher=Dover|year=1988|isbn = 0-486-65812-0}}.
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| | |
| '''Physical treatments'''
| |
| * {{cite book|author=Aris, R.|title=Vectors, Tensors and the Basic Equations of Fluid Mechanics|publisher=Dover|year=1990|isbn=978-0-486-66110-0}}
| |
| * {{cite book|author = [[Richard Feynman|Feynman, R.]], Leighton, R., and Sands, M.|year=2005|edition=2nd ed|title=[[The Feynman Lectures on Physics]], Volume I|publisher=Addison Wesley|isbn=978-0-8053-9046-9|chapter=Chapter 11}}
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| ==External links==
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| {{commons category|Vectors}}
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| {{Wikibooks|Waves|Vectors}}
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| * {{springer|title=Vector|id=p/v096340}}
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| * [http://wwwppd.nrl.navy.mil/nrlformulary/vector_identities.pdf Online vector identities] ([[Portable Document Format|PDF]])
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| * [http://www.marco-learningsystems.com/pages/roche/introvectors.htm Introducing Vectors] A conceptual introduction ([[applied mathematics]])
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| * [http://phy.hk/wiki/englishhtm/Vector.htm Addition of forces (vectors)] Java Applet
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| * [http://www.xna-connection.com/category/Articles/Les-vecteurs French tutorials on vectors and their application to video games]
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| {{linear algebra}}
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| [[Category:Abstract algebra]]
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| [[Category:Vector calculus]]
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| [[Category:Linear algebra]]
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| [[Category:Concepts in physics]]
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| [[Category:Vectors| ]]
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| {{Link GA|fr}}
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| {{Link FA|mk}}
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