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96-405: The book carries the subtitle "A text-book for the use of students of mathematics and physics. Founded upon the lectures of J. Willard Gibbs, Ph.D., LL.D." The first chapter covers vectors in three spatial dimensions, the concept of a (real) scalar , and the product of a scalar with a vector. The second chapter introduces the dot and cross products for pairs of vectors. These are extended to

192-426: A {\displaystyle {\mathfrak {a}}} . Vectors are usually shown in graphs or other diagrams as arrows (directed line segments ), as illustrated in the figure. Here, the point A is called the origin , tail , base , or initial point , and the 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 , while

288-473: A z = a x i + a y j + a z k . {\displaystyle \mathbf {a} =\mathbf {a} _{x}+\mathbf {a} _{y}+\mathbf {a} _{z}=a_{x}{\mathbf {i} }+a_{y}{\mathbf {j} }+a_{z}{\mathbf {k} }.} The notation e i is compatible with the index notation and the summation convention commonly used in higher level mathematics, physics, and engineering. As explained above ,

384-854: A 1 a 2 a 3 ] = [ a 1   a 2   a 3 ] T . {\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\\end{bmatrix}}=[a_{1}\ a_{2}\ a_{3}]^{\operatorname {T} }.} 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: e 1 = ( 1 , 0 , 0 ) ,   e 2 = ( 0 , 1 , 0 ) ,   e 3 = ( 0 , 0 , 1 ) . {\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1).} These have

480-412: A 1 , a 2 , a 3 ) . {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3}).} also written, a = ( a x , a y , a z ) . {\displaystyle \mathbf {a} =(a_{x},a_{y},a_{z}).} This can be generalised to n-dimensional Euclidean space (or R ). a = (

576-397: A 1 , a 2 , a 3 , ⋯ , a n − 1 , a n ) . {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3},\cdots ,a_{n-1},a_{n}).} These numbers are often arranged into a column vector or row vector , particularly when dealing with matrices , as follows: a = [

672-405: A 1 = b 1 , a 2 = b 2 , a 3 = b 3 . {\displaystyle a_{1}=b_{1},\quad a_{2}=b_{2},\quad a_{3}=b_{3}.\,} Two vectors are opposite if they have the same magnitude but opposite direction ; so two vectors Units of measurement A unit of measurement , or unit of measure ,

768-598: A 3 ( 0 , 0 , 1 ) ,   {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3})=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1),\ } or a = a 1 + a 2 + a 3 = a 1 e 1 + a 2 e 2 + a 3 e 3 , {\displaystyle \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},} where

864-721: A 1 , a 2 , a 3 are called the vector components (or vector projections ) of a on the basis vectors or, equivalently, on the corresponding Cartesian axes x , y , and z (see figure), while a 1 , a 2 , a 3 are the respective scalar components (or scalar projections). In introductory physics textbooks, the standard basis vectors are often denoted i , j , k {\displaystyle \mathbf {i} ,\mathbf {j} ,\mathbf {k} } instead (or x ^ , y ^ , z ^ {\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} } , in which

960-412: A . ( Uppercase letters are typically used to represent matrices .) Other conventions include a → {\displaystyle {\vec {a}}} or a , especially in handwriting. Alternatively, some use a tilde (~) or a wavy underline drawn beneath the symbol, e.g. a ∼ {\displaystyle {\underset {^{\sim }}{a}}} , which

1056-418: A Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector ) is a geometric object that has magnitude (or length ) and direction . Euclidean vectors can be added and scaled to form a vector space . A vector quantity is a vector-valued physical quantity , including units of measurement and possibly a support , formulated as a directed line segment . A vector

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1152-483: A barleycorn . A system of measurement is a collection of units of measurement and rules relating them to each other. As science progressed, a need arose to relate the measurement systems of different quantities, like length and weight and volume. The effort of attempting to relate different traditional systems between each other exposed many inconsistencies, and brought about the development of new units and systems. Systems of units vary from country to country. Some of

1248-418: A directed line segment , or arrow, in a Euclidean space . 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 abstract vectors, as they are elements of

1344-484: A linear space in 1844 and 1862, and W. K. Clifford published Elements of Dynamic in 1878, so as Gibbs was teaching physics in the 1880s he took these developments into consideration for his students. A pamphlet that he printed for them acknowledges both Grassmann and Clifford. The influence of Grassmann is seen in the bivectors , and the influence of Clifford in the decomposition of the quaternion product into scalar product and cross product . In 1888 Gibbs sent

1440-440: A scalar triple product and a quadruple product. Pages 77–81 cover the essentials of spherical trigonometry , a topic of considerable interest at the time because of its use in celestial navigation . The third chapter introduces the vector calculus notation based on the del operator . The Helmholtz decomposition of a vector field is given on page 237. The final eight pages develop bivectors as these were integral to

1536-600: A Euclidean space E is defined as a set to which is associated an inner product space of finite dimension over the reals E → , {\displaystyle {\overrightarrow {E}},} and a group action of the additive group of E → , {\displaystyle {\overrightarrow {E}},} which is free and transitive (See Affine space for details of this construction). The elements of E → {\displaystyle {\overrightarrow {E}}} are called translations . It has been proven that

1632-442: A basis does not affect the properties of a vector or its behaviour under transformations. A vector can also be broken up with respect to "non-fixed" basis vectors that change their 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). Moreover, the radial and tangential components of

1728-728: A condition may be emphasized calling the result a bound vector . When only the magnitude and direction of the vector matter, and the particular initial or terminal points are of no importance, the vector is called a free vector . The distinction between bound and free vectors is especially relevant in mechanics, where a force applied to a body has a point of contact (see resultant force and couple ). Two arrows A B ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{AB}}} and A ′ B ′ ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{A'B'}}} in space represent

1824-414: A convenient algebraic characterization of both angle (a function of the dot product between any two non-zero vectors) and length (the square root of the dot product of a vector by itself). In three dimensions, it is further possible to define the cross product , which supplies an algebraic characterization of the area and orientation in space of the parallelogram defined by two vectors (used as sides of

1920-436: A convenient numerical fashion. For example, the sum of the two (free) vectors (1, 2, 3) and (−2, 0, 4) is the (free) vector ( 1 , 2 , 3 ) + ( − 2 , 0 , 4 ) = ( 1 − 2 , 2 + 0 , 3 + 4 ) = ( − 1 , 2 , 7 ) . {\displaystyle (1,2,3)+(-2,0,4)=(1-2,2+0,3+4)=(-1,2,7)\,.} In

2016-537: A copy of his pamphlet to Oliver Heaviside who was formulating his own vectorial system in the Transactions of the Royal Society , praised Gibbs' "little book", saying it "deserves to be well known". However, he also noted that it was "much too condensed for a first introduction to the subject". On the occasion of the bicentennial of Yale University, a series of publications were to be issued to showcase Yale's role in

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2112-425: A corresponding bound vector, in this sense, whose initial point has the coordinates of the origin O = (0, 0, 0) . It is then determined by the coordinates of that bound vector's terminal point. Thus the free vector represented by (1, 0, 0) is a vector of unit length—pointing along the direction of the positive x -axis. This coordinate representation of free vectors allows their algebraic features to be expressed in

2208-567: A new graduate student in mathematics. He had learned about quaternions from James Mills Peirce at Harvard, but Dean A. W. Phillips persuaded him to take Gibbs's course on vectors, which treated similar problems from a rather different perspective. After Wilson had completed the course, Morris approached him about the project of producing a textbook . Wilson wrote the book by expanding his own class notes, providing exercises , and consulting with others (including his father). Vector (geometric) In mathematics , physics , and engineering ,

2304-417: A period of more than 200 years. About a dozen people contributed significantly to its development. In 1835, Giusto Bellavitis abstracted the basic idea when he established the concept of equipollence . Working in a Euclidean plane, he made equipollent any pair of parallel line segments of the same length and orientation. Essentially, he realized an equivalence relation on the pairs of points (bipoints) in

2400-424: A quantity may be described as multiples of that of a familiar entity, which can be easier to contextualize than a value in a formal unit system. For instance, a publication may describe an area in a foreign country as a number of multiples of the area of a region local to the readership. The propensity for certain concepts to be used frequently can give rise to loosely defined "systems" of units. For most quantities

2496-496: A result, units of measure could vary not only from location to location but from person to person. Units not based on the human body could be based on agriculture, as is the case with the furlong and the acre , both based on the amount of land able to be worked by a team of oxen . Metric systems of units have evolved since the adoption of the original metric system in France in 1791. The current international standard metric system

2592-400: A small set of units is required. These units are taken as the base units and the other units are derived units . Thus base units are the units of the quantities which are independent of other quantities and they are the units of length, mass, time, electric current, temperature, luminous intensity and the amount of substance. Derived units are the units of the quantities which are derived from

2688-425: A space with no notion of length or angle. In physics, as well as mathematics, a vector is often identified with a tuple of components, or list of numbers, that act as scalar coefficients for a set of basis vectors . When the basis is transformed, for example by rotation or stretching, then the components of any vector in terms of that basis also transform in an opposite sense. The vector itself has not changed, but

2784-479: A special kind of vector space called Euclidean space . This particular 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. A Euclidean vector may possess a definite initial point and terminal point ; such

2880-602: A subject of governmental regulation, to ensure fairness and transparency. The International Bureau of Weights and Measures (BIPM) is tasked with ensuring worldwide uniformity of measurements and their traceability to the International System of Units (SI). Metrology is the science of developing nationally and internationally accepted units of measurement. In physics and metrology, units are standards for measurement of physical quantities that need clear definitions to be useful. Reproducibility of experimental results

2976-433: A unit is necessary to communicate values of that physical quantity. For example, conveying to someone a particular length without using some sort of unit is impossible, because a length cannot be described without a reference used to make sense of the value given. But not all quantities require a unit of their own. Using physical laws, units of quantities can be expressed as combinations of units of other quantities. Thus only

Vector Analysis - Misplaced Pages Continue

3072-406: A vector is often described by a set of vector components that add up to form the given vector. Typically, these components are the 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. The decomposition or resolution of a vector into components is not unique, because it depends on

3168-726: A vector is one type of tensor . In pure mathematics , a vector is any element of a vector space over some 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". Vectors are usually denoted in lowercase boldface, as in u {\displaystyle \mathbf {u} } , v {\displaystyle \mathbf {v} } and w {\displaystyle \mathbf {w} } , or in lowercase italic boldface, as in

3264-494: A vector pointing into and behind the diagram. These can be thought of as viewing the tip of an arrow head on and viewing the flights of an arrow from the back. 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 vectors 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

3360-848: A vector relate to the radius of rotation of an object. The former is parallel to the radius and the latter is orthogonal to it. 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 ). The following section uses the Cartesian coordinate system with basis vectors e 1 = ( 1 , 0 , 0 ) ,   e 2 = ( 0 , 1 , 0 ) ,   e 3 = ( 0 , 0 , 1 ) {\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1)} and assumes that all vectors have

3456-471: Is Minkowski space (which is important to our understanding of special relativity ). However, it is not always possible or desirable to define the length of a vector. This more general type of spatial vector is the subject of vector spaces (for free vectors) and affine spaces (for bound vectors, as each represented by an ordered pair of "points"). One physical example comes from thermodynamics , where many quantities of interest can be considered vectors in

3552-416: 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 meters would be 4 m or −4 m, depending on its direction, and its magnitude would be 4 m regardless. Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to

3648-417: Is a convention for indicating boldface type. If the vector represents a directed distance or displacement from a point A to a point B (see figure), it can also be denoted as A B ⟶ {\displaystyle {\stackrel {\longrightarrow }{AB}}} or AB . In German literature, it was especially common to represent vectors with small fraktur letters such as

3744-470: Is a definite magnitude of a quantity , defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity . Any other quantity of that kind can be expressed as a multiple of the unit of measurement. For example, a length is a physical quantity . The metre (symbol m) is a unit of length that represents a definite predetermined length. For instance, when referencing "10 metres" (or 10 m), what

3840-510: Is actually meant is 10 times the definite predetermined length called "metre". The definition, agreement, and practical use of units of measurement have played a crucial role in human endeavour from early ages up to the present. A multitude of systems of units used to be very common. Now there is a global standard, the International System of Units (SI), the modern form of the metric system . In trade, weights and measures are often

3936-485: Is central to the scientific method . A standard system of units facilitates this. Scientific systems of units are a refinement of the concept of weights and measures historically developed for commercial purposes. Science , medicine , and engineering often use larger and smaller units of measurement than those used in everyday life. The judicious selection of the units of measurement can aid researchers in problem solving (see, for example, dimensional analysis ). In

Vector Analysis - Misplaced Pages Continue

4032-458: Is essentially the modern system of vector analysis. In 1901, Edwin Bidwell Wilson published Vector Analysis , adapted from Gibbs's lectures, which banished any mention of quaternions in the development of vector calculus. In physics and engineering , a vector is typically regarded as a geometric entity characterized by a magnitude and a relative direction . It is formally defined as

4128-409: Is expressed as the product of a numerical value { Z } (a pure number) and a unit [ Z ]: For example, let Z {\displaystyle Z} be "2 metres"; then, { Z } = 2 {\displaystyle \{Z\}=2} is the numerical value and [ Z ] = m e t r e {\displaystyle [Z]=\mathrm {metre} } is the unit. Conversely,

4224-495: Is frequently depicted graphically as an arrow connecting an initial point A with a terminal point B , and denoted by A B ⟶ . {\textstyle {\stackrel {\longrightarrow }{AB}}.} A vector is what is needed to "carry" the point A to the point B ; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planetary revolution around

4320-464: Is now defined as exactly 0.0254  m , and the US and imperial avoirdupois pound is now defined as exactly 0.453 592 37   kg . While the above systems of units are based on arbitrary unit values, formalised as standards, natural units in physics are based on physical principle or are selected to make physical equations easier to work with. For example, atomic units (au) were designed to simplify

4416-431: Is often presented as the standard Euclidean space of dimension n . This is motivated by the fact that every Euclidean space of dimension n is isomorphic to the Euclidean space R n . {\displaystyle \mathbb {R} ^{n}.} More precisely, given such a Euclidean space, one may choose any point O as an origin . By Gram–Schmidt process , one may also find an orthonormal basis of

4512-480: Is the International System of Units (abbreviated to SI). An important feature of modern systems is standardization . Each unit has a universally recognized size. Both the imperial units and US customary units derive from earlier English units . Imperial units were mostly used in the British Commonwealth and the former British Empire . US customary units are still the main system of measurement used in

4608-503: Is the conversion of the unit of measurement in which a quantity is expressed, typically through a multiplicative conversion factor that changes the unit without changing the quantity. This is also often loosely taken to include replacement of a quantity with a corresponding quantity that describes the same physical property. One example of the importance of agreed units is the failure of the NASA Mars Climate Orbiter , which

4704-568: Is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment ( A , B ) ) and same direction (e.g., the direction from A to B ). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars , which have no direction. For example, velocity , forces and acceleration are represented by vectors. In modern geometry, Euclidean spaces are often defined from linear algebra . More precisely,

4800-413: The n -tuple of its Cartesian coordinates, and every vector to its coordinate vector . Since the physicist's concept of force has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force F of 15 newtons . If the positive axis is also directed rightward, then F is represented by the vector 15 N, and if positive points leftward, then the vector for F

4896-856: The 4th and 3rd millennia BC among the ancient peoples of Mesopotamia , Egypt and the Indus Valley , and perhaps also Elam in Persia as well. Weights and measures are mentioned in the Bible (Leviticus 19:35–36). It is a commandment to be honest and have fair measures. In the Magna Carta of 1215 (The Great Charter) with the seal of King John , put before him by the Barons of England, King John agreed in Clause 35 "There shall be one measure of wine throughout our whole realm, and one measure of ale and one measure of corn—namely,

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4992-477: The United States outside of science, medicine, many sectors of industry, and some of government and military, and despite Congress having legally authorised metric measure on 28 July 1866. Some steps towards US metrication have been made, particularly the redefinition of basic US and imperial units to derive exactly from SI units. Since the international yard and pound agreement of 1959 the US and imperial inch

5088-487: The coordinates of the endpoint of the vector, with respect to a given Cartesian coordinate system , and are typically called the scalar components (or scalar projections ) of the vector on the axes of the coordinate system. 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 a = ( 2 , 3 ) . {\displaystyle \mathbf {a} =(2,3).} The notion that

5184-417: The dot product . This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, R n {\displaystyle \mathbb {R} ^{n}} is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product. The Euclidean space R n {\displaystyle \mathbb {R} ^{n}}

5280-417: The 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, are angular displacement and electric current. Consequently, these are not vectors. In the Cartesian coordinate system , a bound vector can be represented by identifying

5376-396: The hat symbol ^ {\displaystyle \mathbf {\hat {}} } typically denotes unit vectors ). In this case, the scalar and vector components are denoted respectively a x , a y , a z , and a x , a y , a z (note the difference in boldface). Thus, a = a x + a y +

5472-401: The metric system , the imperial system , and United States customary units . Historically many of the systems of measurement which had been in use were to some extent based on the dimensions of the human body. Such units, which may be called anthropic units , include the cubit , based on the length of the forearm; the pace , based on the length of a stride; and the foot and hand . As

5568-448: The real line , Hamilton considered the vector v to be the imaginary part of a quaternion: 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. Several other mathematicians developed vector-like systems in

5664-553: The social sciences , there are no standard units of measurement. A unit of measurement is a standardized quantity of a physical property, used as a factor to express occurring quantities of that property. Units of measurement were among the earliest tools invented by humans. Primitive societies needed rudimentary measures for many tasks: constructing dwellings of an appropriate size and shape, fashioning clothing, or bartering food or raw materials. The earliest known uniform systems of measurement seem to have all been created sometime in

5760-491: The 19th century) as equivalence classes under equipollence , of ordered pairs of points; two pairs ( A , B ) and ( C , D ) being equipollent if the points A , B , D , C , in this order, form a parallelogram . Such an equivalence class is called a vector , more precisely, a Euclidean vector. The equivalence class of ( A , B ) is often denoted A B → . {\displaystyle {\overrightarrow {AB}}.} A Euclidean vector

5856-603: The London quart;—and one width of dyed and russet and hauberk cloths—namely, two ells below the selvage..." As of the 21st century, the International System is predominantly used in the world. There exist other unit systems which are used in many places such as the United States Customary System and the Imperial System. The United States is the only industrialized country that has not yet at least mostly converted to

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5952-540: The Sun. 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 . These operations and associated laws qualify Euclidean vectors as an example of

6048-532: The advancement of knowledge. Gibbs was authoring Elementary Principles in Statistical Mechanics for that series. Mindful of the demand for innovative university textbooks, the editor of the series, Professor Morris, wished to include also a volume dedicated to Gibbs's lectures on vectors, but Gibbs's time and attention were entirely absorbed by the Statistical Mechanics . E. B. Wilson was then

6144-534: The associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines Cartesian coordinates of any point P of the space, as the coordinates on this basis of the vector O P → . {\displaystyle {\overrightarrow {OP}}.} These choices define an isomorphism of the given Euclidean space onto R n , {\displaystyle \mathbb {R} ^{n},} by mapping any point to

6240-414: The base quantities and some of the derived units are the units of speed, work, acceleration, energy, pressure etc. Different systems of units are based on different choices of a set of related units including fundamental and derived units. Following ISO 80000-1 , any value or magnitude of a physical quantity is expressed as a comparison to a unit of that quantity. The value of a physical quantity Z

6336-433: The basis has, so the components of the vector 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 the basis. 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

6432-442: The choice of the axes on which the vector is projected. Moreover, the use of Cartesian unit vectors such as x ^ , y ^ , z ^ {\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} } as a basis in which to represent a vector is not mandated. Vectors can also be expressed in terms of an arbitrary basis, including

6528-441: The complete quaternion product. This approach made vector calculations available to engineers—and others working in three dimensions and skeptical of the fourth. 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

6624-410: The coordinates of its initial and terminal point. For instance, the points A = (1, 0, 0) and B = (0, 1, 0) in space determine the bound vector A B → {\displaystyle {\overrightarrow {AB}}} pointing from the point x = 1 on the x -axis to the point y = 1 on the y -axis. In Cartesian coordinates, a free vector may be thought of in terms of

6720-410: The course on the electromagnetic theory of light that Professor Gibbs taught at Yale. First Wilson associates a bivector with an ellipse. The product of the bivector with a complex number on the unit circle is then called an elliptical rotation . Wilson continues with a description of elliptic harmonic motion and the case of stationary waves . Hermann Grassmann had introduced basic ideas of

6816-500: The crew confusing tower instructions (in metres) and altimeter readings (in feet). Three crew and five people on the ground were killed. Thirty-seven were injured. In 1983, a Boeing 767 (which thanks to its pilot's gliding skills landed safely and became known as the Gimli Glider ) ran out of fuel in mid-flight because of two mistakes in figuring the fuel supply of Air Canada 's first aircraft to use metric measurements. This accident

6912-468: The different systems include the centimetre–gram–second , foot–pound–second , metre–kilogram–second systems, and the International System of Units , SI. Among the different systems of units used in the world, the most widely used and internationally accepted one is SI. The base SI units are the second, metre, kilogram, ampere, kelvin, mole and candela; all other SI units are derived from these base units. Systems of measurement in modern use include

7008-433: The direction in which the arrow points indicates the vector's direction. On a two-dimensional diagram, a vector perpendicular to the plane of the diagram is sometimes 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

7104-411: The geometrical and physical settings, it is sometimes possible to associate, in a natural way, a length or magnitude and a direction to vectors. In addition, the notion of direction is strictly associated with the notion of an angle between two vectors. If the dot product of two vectors is defined—a scalar-valued product of two vectors—then it is also possible to define a length; the dot product gives

7200-435: 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 R can be expressed in the form: a = ( a 1 , a 2 , a 3 ) = a 1 ( 1 , 0 , 0 ) + a 2 ( 0 , 1 , 0 ) +

7296-406: 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 pseudovectors and tensors . The vector concept, as it is known today, is the result of a gradual development over

7392-608: The metric system. The systematic effort to develop a universally acceptable system of units dates back to 1790 when the French National Assembly charged the French Academy of Sciences to come up such a unit system. This system was the precursor to the metric system which was quickly developed in France but did not take on universal acceptance until 1875 when The Metric Convention Treaty was signed by 17 nations. After this treaty

7488-513: The middle of the nineteenth century, including Augustin Cauchy , Hermann Grassmann , August Möbius , Comte de Saint-Venant , and 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 that is similar to today's system, and had ideas corresponding to the cross product, scalar product and vector differentiation. Grassmann's work

7584-483: The more generalized concept of vectors defined simply as elements of a vector space . Vectors play an important role in physics : the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement ), their magnitude and direction can still be represented by

7680-425: The numerical value expressed in an arbitrary unit can be obtained as: Units can only be added or subtracted if they are the same type; however units can always be multiplied or divided, as George Gamow used to explain. Let Z {\displaystyle Z} be "2 metres" and W {\displaystyle W} "3 seconds", then There are certain rules that apply to units: Conversion of units

7776-405: The origin as a common base point. A vector a will be written as a = a 1 e 1 + a 2 e 2 + a 3 e 3 . {\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}.} Two vectors are said to be equal if they have

7872-488: The other hand, have units of one-over-distance such as gradient . If you change units (a special case of a change of basis ) from meters to millimeters, 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  K /m becomes 0.001 K/mm—a covariant change in value (for more, see covariance and contravariance of vectors ). Tensors are another type of quantity that behave in this way;

7968-408: The parallelogram). In any dimension (and, in particular, higher dimensions), it is possible to define the exterior product , which (among other things) supplies an algebraic characterization of the area and orientation in space of the n -dimensional parallelotope defined by n vectors. In a pseudo-Euclidean space , a vector's squared length can be positive, negative, or zero. An important example

8064-434: The plane, and thus erected the first space of vectors in the plane. The term vector was introduced by William Rowan Hamilton as part of a quaternion , which is a sum q = s + v of a real number s (also called scalar ) and a 3-dimensional vector . Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments. As complex numbers use an imaginary unit to complement

8160-596: The rules of vector addition. An example is velocity , the magnitude of which is speed . For instance, 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 , linear acceleration, angular acceleration , linear momentum , and angular momentum . Other physical vectors, such as

8256-563: The same free vector if they have the same magnitude and direction: that is, they are equipollent if the quadrilateral ABB′A′ is a parallelogram . If the Euclidean space is equipped with a choice of origin , then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin. The term vector also has generalizations to higher dimensions, and to more formal approaches with much wider applications. In classical Euclidean geometry (i.e., synthetic geometry ), vectors were introduced (during

8352-699: The same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors a = a 1 e 1 + a 2 e 2 + a 3 e 3 {\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}} and b = b 1 e 1 + b 2 e 2 + b 3 e 3 {\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}} are equal if

8448-434: The same unit for the distance between two cities and the length of a needle. Thus, historically they would develop independently. One way to make large numbers or small fractions easier to read, is to use unit prefixes . At some point in time though, the need to relate the two units might arise, and consequently the need to choose one unit as defining the other or vice versa. For example, an inch could be defined in terms of

8544-412: The tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation O A → {\displaystyle {\overrightarrow {OA}}} is usually deemed not necessary (and is indeed rarely used). In three dimensional Euclidean space (or R ), vectors are identified with triples of scalar components: a = (

8640-487: The two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations. Sometimes, Euclidean vectors are considered without reference to a Euclidean space. In this case, a Euclidean vector is an element of a normed vector space of finite dimension over the reals, or, typically, an element of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} equipped with

8736-702: The unit vectors of a cylindrical coordinate system ( ρ ^ , ϕ ^ , z ^ {\displaystyle {\boldsymbol {\hat {\rho }}},{\boldsymbol {\hat {\phi }}},\mathbf {\hat {z}} } ) or spherical coordinate system ( r ^ , θ ^ , ϕ ^ {\displaystyle \mathbf {\hat {r}} ,{\boldsymbol {\hat {\theta }}},{\boldsymbol {\hat {\phi }}}} ). The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry, respectively. The choice of

8832-515: The wave equation in atomic physics . Some unusual and non-standard units may be encountered in sciences. These may include the solar mass ( 2 × 10  kg ), the megaton (the energy released by detonating one million tons of trinitrotoluene , TNT) and the electronvolt . To reduce the incidence of retail fraud, many national statutes have standard definitions of weights and measures that may be used (hence " statute measure "), and these are verified by legal officers. In informal settings,

8928-453: Was accidentally destroyed on a mission to Mars in September 1999 (instead of entering orbit) due to miscommunications about the value of forces: different computer programs used different units of measurement ( newton versus pound force ). Considerable amounts of effort, time, and money were wasted. On 15 April 1999, Korean Air cargo flight 6316 from Shanghai to Seoul was lost due to

9024-407: Was largely neglected until the 1870s. Peter Guthrie Tait carried the quaternion standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇. 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

9120-575: Was signed, a General Conference of Weights and Measures (CGPM) was established. The CGPM produced the current SI, which was adopted in 1954 at the 10th Conference of Weights and Measures. Currently, the United States is a dual-system society which uses both the SI and the US Customary system. The use of a single unit of measurement for some quantity has obvious drawbacks. For example, it is impractical to use

9216-549: Was the result of both confusion due to the simultaneous use of metric and Imperial measures and confusion of mass and volume measures. When planning his journey across the Atlantic Ocean in the 1480s, Columbus mistakenly assumed that the mile referred to in the Arabic estimate of ⁠56 + 2 / 3 ⁠ miles for the size of a degree was the same as the actually much shorter Italian mile of 1,480 metres. His estimate for

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