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64-432: Velocity is the speed in combination with the direction of motion of an object . Velocity is a fundamental concept in kinematics , the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity : both magnitude and direction are needed to define it. The scalar absolute value ( magnitude ) of velocity is called speed , being a coherent derived unit whose quantity

128-496: A ⋅ x ) {\displaystyle \therefore v^{2}=u^{2}+2({\boldsymbol {a}}\cdot {\boldsymbol {x}})} where v = | v | etc. The above equations are valid for both Newtonian mechanics and special relativity . Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on

192-605: A ) ⋅ x = ( 2 a ) ⋅ ( u t + 1 2 a t 2 ) = 2 t ( a ⋅ u ) + a 2 t 2 = v 2 − u 2 {\displaystyle (2{\boldsymbol {a}})\cdot {\boldsymbol {x}}=(2{\boldsymbol {a}})\cdot ({\boldsymbol {u}}t+{\tfrac {1}{2}}{\boldsymbol {a}}t^{2})=2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}=v^{2}-u^{2}} ∴ v 2 = u 2 + 2 (

256-447: A = d v d t . {\displaystyle {\boldsymbol {a}}={\frac {d{\boldsymbol {v}}}{dt}}.} From there, velocity is expressed as the area under an a ( t ) acceleration vs. time graph. As above, this is done using the concept of the integral: v = ∫ a   d t . {\displaystyle {\boldsymbol {v}}=\int {\boldsymbol {a}}\ dt.} In

320-455: A 1 , a 2 , ..., a n ) where a i ∈ K and n is the dimension of the vector space in consideration.). For example, every real vector space of dimension n is isomorphic to the n -dimensional real space R . Alternatively, a vector space V can be equipped with a norm function that assigns to every vector v in V a scalar || v ||. By definition, multiplying v by a scalar k also multiplies its norm by | k |. If || v ||

384-525: A very short period of time, is called instantaneous speed . By looking at a speedometer , one can read the instantaneous speed of a car at any instant. A car travelling at 50 km/h generally goes for less than one hour at a constant speed, but if it did go at that speed for a full hour, it would travel 50 km. If the vehicle continued at that speed for half an hour, it would cover half that distance (25 km). If it continued for only one minute, it would cover about 833 m. In mathematical terms,

448-489: A (linear) function space , kf is the function x ↦ k ( f ( x )) . The scalars can be taken from any field, including the rational , algebraic , real, and complex numbers, as well as finite fields . According to a fundamental theorem of linear algebra, every vector space has a basis . It follows that every vector space over a field K is isomorphic to the corresponding coordinate vector space where each coordinate consists of elements of K (E.g., coordinates (

512-413: A 4-hour trip, the distance covered is found to be 320 kilometres. Expressed in graphical language, the slope of a tangent line at any point of a distance-time graph is the instantaneous speed at this point, while the slope of a chord line of the same graph is the average speed during the time interval covered by the chord. Average speed of an object is Vav = s÷t Speed denotes only how fast an object

576-417: A constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes. Hence, the car is considered to be undergoing an acceleration. Since the derivative of the position with respect to time gives the change in position (in metres ) divided by the change in time (in seconds ), velocity is measured in metres per second (m/s). Velocity

640-520: A scalar multiplication operation that takes a scalar k and a vector v to form another vector k v . For example, in a coordinate space , the scalar multiplication k ( v 1 , v 2 , … , v n ) {\displaystyle k(v_{1},v_{2},\dots ,v_{n})} yields ( k v 1 , k v 2 , … , k v n ) {\displaystyle (kv_{1},kv_{2},\dots ,kv_{n})} . In

704-442: A scalar product is called an inner product space . A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector . The term scalar is also sometimes used informally to mean a vector, matrix , tensor , or other, usually, "compound" value that is actually reduced to a single component. Thus, for example, the product of a 1 ×  n matrix and an n  × 1 matrix, which

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768-417: A straight line), this can be simplified to v = s / t {\displaystyle v=s/t} . The average speed over a finite time interval is the total distance travelled divided by the time duration. Different from instantaneous speed, average speed is defined as the total distance covered divided by the time interval. For example, if a distance of 80 kilometres is driven in 1 hour,

832-587: A two-dimensional system, where there is an x-axis and a y-axis, corresponding velocity components are defined as v x = d x / d t , {\displaystyle v_{x}=dx/dt,} v y = d y / d t . {\displaystyle v_{y}=dy/dt.} The two-dimensional velocity vector is then defined as v =< v x , v y > {\displaystyle {\textbf {v}}=<v_{x},v_{y}>} . The magnitude of this vector represents speed and

896-399: A velocity vector, denotes only how fast an object is moving, while velocity indicates both an object's speed and direction. To have a constant velocity , an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at

960-468: Is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. The drag force, F D {\displaystyle F_{D}} , is dependent on the square of velocity and is given as F D = 1 2 ρ v 2 C D A {\displaystyle F_{D}\,=\,{\tfrac {1}{2}}\,\rho \,v^{2}\,C_{D}\,A} where Escape velocity

1024-417: Is always less than or equal to the average speed of an object. This can be seen by realizing that while distance is always strictly increasing, displacement can increase or decrease in magnitude as well as change direction. In terms of a displacement-time ( x vs. t ) graph, the instantaneous velocity (or, simply, velocity) can be thought of as the slope of the tangent line to the curve at any point , and

1088-720: Is defined as v =< v x , v y , v z > {\displaystyle {\textbf {v}}=<v_{x},v_{y},v_{z}>} with its magnitude also representing speed and being determined by | v | = v x 2 + v y 2 + v z 2 . {\displaystyle |v|={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}.} While some textbooks use subscript notation to define Cartesian components of velocity, others use u {\displaystyle u} , v {\displaystyle v} , and w {\displaystyle w} for

1152-499: Is defined as the rate of change of position with respect to time, which may also be referred to as the instantaneous velocity to emphasize the distinction from the average velocity. In some applications the average velocity of an object might be needed, that is to say, the constant velocity that would provide the same resultant displacement as a variable velocity in the same time interval, v ( t ) , over some time period Δ t . Average velocity can be calculated as: The average velocity

1216-435: Is defined as the rate of change of position, it is often common to start with an expression for an object's acceleration . As seen by the three green tangent lines in the figure, an object's instantaneous acceleration at a point in time is the slope of the line tangent to the curve of a v ( t ) graph at that point. In other words, instantaneous acceleration is defined as the derivative of velocity with respect to time:

1280-565: Is formally a 1 × 1 matrix, is often said to be a scalar . The real component of a quaternion is also called its scalar part . The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix . The word scalar derives from the Latin word scalaris , an adjectival form of scala (Latin for "ladder"), from which the English word scale also comes. The first recorded usage of

1344-483: Is found by the distance formula as | v | = v x 2 + v y 2 . {\displaystyle |v|={\sqrt {v_{x}^{2}+v_{y}^{2}}}.} In three-dimensional systems where there is an additional z-axis, the corresponding velocity component is defined as v z = d z / d t . {\displaystyle v_{z}=dz/dt.} The three-dimensional velocity vector

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1408-634: Is given by the harmonic mean of the speeds v ¯ = n ( 1 v 1 + 1 v 2 + 1 v 3 + ⋯ + 1 v n ) − 1 = n ( ∑ i = 1 n 1 v i ) − 1 . {\displaystyle {\bar {v}}=n\left({1 \over v_{1}}+{1 \over v_{2}}+{1 \over v_{3}}+\dots +{1 \over v_{n}}\right)^{-1}=n\left(\sum _{i=1}^{n}{\frac {1}{v_{i}}}\right)^{-1}.} Although velocity

1472-443: Is interpreted as the length of v , this operation can be described as scaling the length of v by k . A vector space equipped with a norm is called a normed vector space (or normed linear space ). The norm is usually defined to be an element of V 's scalar field K , which restricts the latter to fields that support the notion of sign. Moreover, if V has dimension 2 or more, K must be closed under square root, as well as

1536-405: Is judged to be more rapid than another when at a given moment the first object is behind and a moment or so later ahead of the other object." Scalar (mathematics) A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with

1600-469: Is known as moment of inertia . If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitational orbit , angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion . Speed In kinematics ,

1664-500: Is measured in the SI ( metric system ) as metres per second (m/s or m⋅s). For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector. If there is a change in speed, direction or both, then the object is said to be undergoing an acceleration . The average velocity of an object over a period of time is its change in position , Δ s {\displaystyle \Delta s} , divided by

1728-460: Is moving, whereas velocity describes both how fast and in which direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified. The big difference can be discerned when considering movement around a circle . When something moves in a circular path and returns to its starting point, its average velocity

1792-474: Is position and r ^ {\displaystyle {\hat {\boldsymbol {r}}}} is the radial direction. The transverse speed (or magnitude of the transverse velocity) is the magnitude of the cross product of the unit vector in the radial direction and the velocity vector. It is also the dot product of velocity and transverse direction, or the product of the angular speed ω {\displaystyle \omega } and

1856-436: Is speed, d {\displaystyle d} is distance, and t {\displaystyle t} is time. A cyclist who covers 30 metres in a time of 2 seconds, for example, has a speed of 15 metres per second. Objects in motion often have variations in speed (a car might travel along a street at 50 km/h, slow to 0 km/h, and then reach 30 km/h). Speed at some instant, or assumed constant during

1920-409: Is the gravitational constant and g is the gravitational acceleration . The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. This makes "escape velocity" somewhat of a misnomer, as the more correct term would be "escape speed": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of

1984-769: Is the component of velocity along a circle centered at the origin. v = v T + v R {\displaystyle {\boldsymbol {v}}={\boldsymbol {v}}_{T}+{\boldsymbol {v}}_{R}} where The radial speed (or magnitude of the radial velocity) is the dot product of the velocity vector and the unit vector in the radial direction. v R = v ⋅ r | r | = v ⋅ r ^ {\displaystyle v_{R}={\frac {{\boldsymbol {v}}\cdot {\boldsymbol {r}}}{\left|{\boldsymbol {r}}\right|}}={\boldsymbol {v}}\cdot {\hat {\boldsymbol {r}}}} where r {\displaystyle {\boldsymbol {r}}}

Velocity - Misplaced Pages Continue

2048-400: Is the length of the path (also known as the distance) travelled until time t {\displaystyle t} , the speed equals the time derivative of s {\displaystyle s} : v = d s d t . {\displaystyle v={\frac {ds}{dt}}.} In the special case where the velocity is constant (that is, constant speed in

2112-417: Is the mass of the object. The kinetic energy of a moving object is dependent on its velocity and is given by the equation E k = 1 2 m v 2 {\displaystyle E_{\text{k}}={\tfrac {1}{2}}mv^{2}} where E k is the kinetic energy. Kinetic energy is a scalar quantity as it depends on the square of the velocity. In fluid dynamics , drag

2176-461: Is the mass times the distance to the origin times the transverse velocity, or equivalently, the mass times the distance squared times the angular speed. The sign convention for angular momentum is the same as that for angular velocity. L = m r v T = m r 2 ω {\displaystyle L=mrv_{T}=mr^{2}\omega } where The expression m r 2 {\displaystyle mr^{2}}

2240-546: Is the minimum speed a ballistic object needs to escape from a massive body such as Earth. It represents the kinetic energy that, when added to the object's gravitational potential energy (which is always negative), is equal to zero. The general formula for the escape velocity of an object at a distance r from the center of a planet with mass M is v e = 2 G M r = 2 g r , {\displaystyle v_{\text{e}}={\sqrt {\frac {2GM}{r}}}={\sqrt {2gr}},} where G

2304-463: Is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system). The radial and traverse velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. The transverse velocity

2368-453: Is the speed of light. Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics deal with the relative motion of two or more particles. Consider an object A moving with velocity vector v and an object B with velocity vector w ; these absolute velocities are typically expressed in

2432-420: Is zero, but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle. This is because the average velocity is calculated by considering only the displacement between the starting and end points, whereas the average speed considers only the total distance travelled. Units of speed include: (* = approximate values) According to Jean Piaget ,

2496-433: The x {\displaystyle x} -, y {\displaystyle y} -, and z {\displaystyle z} -axes respectively. In polar coordinates , a two-dimensional velocity is described by a radial velocity , defined as the component of velocity away from or toward the origin, and a transverse velocity , perpendicular to the radial one. Both arise from angular velocity , which

2560-423: The derivative of the position with respect to time: v = lim Δ t → 0 Δ s Δ t = d s d t . {\displaystyle {\boldsymbol {v}}=\lim _{{\Delta t}\to 0}{\frac {\Delta {\boldsymbol {s}}}{\Delta t}}={\frac {d{\boldsymbol {s}}}{dt}}.} From this derivative equation, in

2624-399: The limit of the average speed as the duration of the time interval approaches zero. Speed is the magnitude of velocity (a vector), which indicates additionally the direction of motion. Speed has the dimensions of distance divided by time. The SI unit of speed is the metre per second (m/s), but the most common unit of speed in everyday usage is the kilometre per hour (km/h) or, in

Velocity - Misplaced Pages Continue

2688-399: The speed (commonly referred to as v ) of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a non-negative scalar quantity. The average speed of an object in an interval of time is the distance travelled by the object divided by the duration of the interval; the instantaneous speed is

2752-572: The US and the UK, miles per hour (mph). For air and marine travel, the knot is commonly used. The fastest possible speed at which energy or information can travel, according to special relativity , is the speed of light in vacuum c = 299 792 458 metres per second (approximately 1 079 000 000  km/h or 671 000 000  mph ). Matter cannot quite reach the speed of light, as this would require an infinite amount of energy. In relativity physics,

2816-423: The average speed is 80 kilometres per hour. Likewise, if 320 kilometres are travelled in 4 hours, the average speed is also 80 kilometres per hour. When a distance in kilometres (km) is divided by a time in hours (h), the result is in kilometres per hour (km/h). Average speed does not describe the speed variations that may have taken place during shorter time intervals (as it is the entire distance covered divided by

2880-1574: The average velocity as the slope of the secant line between two points with t coordinates equal to the boundaries of the time period for the average velocity. If t 1 = t 2 = t 3 = ... = t , then average speed is given by the arithmetic mean of the speeds v ¯ = v 1 + v 2 + v 3 + ⋯ + v n n = 1 n ∑ i = 1 n v i {\displaystyle {\bar {v}}={v_{1}+v_{2}+v_{3}+\dots +v_{n} \over n}={\frac {1}{n}}\sum _{i=1}^{n}{v_{i}}} v ¯ = s 1 + s 2 + s 3 + ⋯ + s n t 1 + t 2 + t 3 + ⋯ + t n = s 1 + s 2 + s 3 + ⋯ + s n s 1 v 1 + s 2 v 2 + s 3 v 3 + ⋯ + s n v n {\displaystyle {\bar {v}}={s_{1}+s_{2}+s_{3}+\dots +s_{n} \over t_{1}+t_{2}+t_{3}+\dots +t_{n}}={{s_{1}+s_{2}+s_{3}+\dots +s_{n}} \over {{s_{1} \over v_{1}}+{s_{2} \over v_{2}}+{s_{3} \over v_{3}}+\dots +{s_{n} \over v_{n}}}}} If s 1 = s 2 = s 3 = ... = s , then average speed

2944-455: The base body as long as it does not intersect with something in its path. In special relativity , the dimensionless Lorentz factor appears frequently, and is given by γ = 1 1 − v 2 c 2 {\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}} where γ is the Lorentz factor and c

3008-445: The concept of rapidity replaces the classical idea of speed. Italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time. In equation form, that is v = d t , {\displaystyle v={\frac {d}{t}},} where v {\displaystyle v}

3072-402: The concept of an instantaneous velocity might at first seem counter-intuitive, it may be thought of as the velocity that the object would continue to travel at if it stopped accelerating at that moment. While the terms speed and velocity are often colloquially used interchangeably to connote how fast an object is moving, in scientific terms they are different. Speed, the scalar magnitude of

3136-457: The duration of the period, Δ t {\displaystyle \Delta t} , given mathematically as v ¯ = Δ s Δ t . {\displaystyle {\bar {v}}={\frac {\Delta s}{\Delta t}}.} The instantaneous velocity of an object is the limit average velocity as the time interval approaches zero. At any particular time t , it can be calculated as

3200-402: The four arithmetic operations; thus the rational numbers Q are excluded, but the surd field is acceptable. For this reason, not every scalar product space is a normed vector space. When the requirement that the set of scalars form a field is relaxed so that it need only form a ring (so that, for example, the division of scalars need not be defined, or the scalars need not be commutative ),

3264-539: The inertial frame chosen is that in which the latter of the two mentioned objects is in rest. In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with special relativity in which velocities depend on the choice of reference frame. In the one-dimensional case, the velocities are scalars and the equation is either: v rel = v − ( − w ) , {\displaystyle v_{\text{rel}}=v-(-w),} if

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3328-679: The instantaneous speed v {\displaystyle v} is defined as the magnitude of the instantaneous velocity v {\displaystyle {\boldsymbol {v}}} , that is, the derivative of the position r {\displaystyle {\boldsymbol {r}}} with respect to time : v = | v | = | r ˙ | = | d r d t | . {\displaystyle v=\left|{\boldsymbol {v}}\right|=\left|{\dot {\boldsymbol {r}}}\right|=\left|{\frac {d{\boldsymbol {r}}}{dt}}\right|\,.} If s {\displaystyle s}

3392-434: The intuition for the notion of speed in humans precedes that of duration, and is based on the notion of outdistancing. Piaget studied this subject inspired by a question asked to him in 1928 by Albert Einstein : "In what order do children acquire the concepts of time and speed?" Children's early concept of speed is based on "overtaking", taking only temporal and spatial orders into consideration, specifically: "A moving object

3456-477: The one-dimensional case it can be seen that the area under a velocity vs. time ( v vs. t graph) is the displacement, s . In calculus terms, the integral of the velocity function v ( t ) is the displacement function s ( t ) . In the figure, this corresponds to the yellow area under the curve. s = ∫ v   d t . {\displaystyle {\boldsymbol {s}}=\int {\boldsymbol {v}}\ dt.} Although

3520-754: The radius (the magnitude of the position). v T = | r × v | | r | = v ⋅ t ^ = ω | r | {\displaystyle v_{T}={\frac {|{\boldsymbol {r}}\times {\boldsymbol {v}}|}{|{\boldsymbol {r}}|}}={\boldsymbol {v}}\cdot {\hat {\boldsymbol {t}}}=\omega |{\boldsymbol {r}}|} such that ω = | r × v | | r | 2 . {\displaystyle \omega ={\frac {|{\boldsymbol {r}}\times {\boldsymbol {v}}|}{|{\boldsymbol {r}}|^{2}}}.} Angular momentum in scalar form

3584-406: The resulting more general algebraic structure is called a module . In this case the "scalars" may be complicated objects. For instance, if R is a ring, the vectors of the product space R can be made into a module with the n × n matrices with entries from R as the scalars. Another example comes from manifold theory , where the space of sections of the tangent bundle forms a module over

3648-710: The same inertial reference frame . Then, the velocity of object A relative to object B is defined as the difference of the two velocity vectors: v A  relative to  B = v − w {\displaystyle {\boldsymbol {v}}_{A{\text{ relative to }}B}={\boldsymbol {v}}-{\boldsymbol {w}}} Similarly, the relative velocity of object B moving with velocity w , relative to object A moving with velocity v is: v B  relative to  A = w − v {\displaystyle {\boldsymbol {v}}_{B{\text{ relative to }}A}={\boldsymbol {w}}-{\boldsymbol {v}}} Usually,

3712-432: The special case of constant acceleration, velocity can be studied using the suvat equations . By considering a as being equal to some arbitrary constant vector, this shows v = u + a t {\displaystyle {\boldsymbol {v}}={\boldsymbol {u}}+{\boldsymbol {a}}t} with v as the velocity at time t and u as the velocity at time t = 0 . By combining this equation with

3776-414: The suvat equation x = u t + a t /2 , it is possible to relate the displacement and the average velocity by x = ( u + v ) 2 t = v ¯ t . {\displaystyle {\boldsymbol {x}}={\frac {({\boldsymbol {u}}+{\boldsymbol {v}})}{2}}t={\boldsymbol {\bar {v}}}t.} It is also possible to derive an expression for

3840-432: The total time of travel), and so average speed is often quite different from a value of instantaneous speed. If the average speed and the time of travel are known, the distance travelled can be calculated by rearranging the definition to d = v ¯ t . {\displaystyle d={\boldsymbol {\bar {v}}}t\,.} Using this equation for an average speed of 80 kilometres per hour on

3904-401: The two objects are moving in opposite directions, or: v rel = v − ( + w ) , {\displaystyle v_{\text{rel}}=v-(+w),} if the two objects are moving in the same direction. In multi-dimensional Cartesian coordinate systems , velocity is broken up into components that correspond with each dimensional axis of the coordinate system. In

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3968-555: The value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words, only relative velocity can be calculated. In classical mechanics, Newton's second law defines momentum , p, as a vector that is the product of an object's mass and velocity, given mathematically as p = m v {\displaystyle {\boldsymbol {p}}=m{\boldsymbol {v}}} where m

4032-654: The velocity independent of time, known as the Torricelli equation , as follows: v 2 = v ⋅ v = ( u + a t ) ⋅ ( u + a t ) = u 2 + 2 t ( a ⋅ u ) + a 2 t 2 {\displaystyle v^{2}={\boldsymbol {v}}\cdot {\boldsymbol {v}}=({\boldsymbol {u}}+{\boldsymbol {a}}t)\cdot ({\boldsymbol {u}}+{\boldsymbol {a}}t)=u^{2}+2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}} ( 2

4096-568: The word "scalar" in mathematics occurs in François Viète 's Analytic Art ( In artem analyticem isagoge ) (1591): According to a citation in the Oxford English Dictionary the first recorded usage of the term "scalar" in English came with W. R. Hamilton in 1846, referring to the real part of a quaternion: A vector space is defined as a set of vectors (additive abelian group ), a set of scalars ( field ), and

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