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This article is about vectors that have a particular relation to the spatial coordinates.For a generalization, see vector space.For other uses, see vector.The magnitude is the length of the segment and the direction characterizes the displacement of B relative to A: how much one should move the point A to "carry" it to the point B.Many algebraic operations on real numbers have close analogues for vectors.Vectors can be added, subtracted, multiplied by a number, and flipped around so that the direction is reversed.These operations obey the familiar algebraic laws: commutativity, associativity, distributivity.Cartesian coordinates provide a systematic way of describing vectors and operations on them.Addition of vectors and multiplication of a vector by a scalar are simply done component by component, see coordinate vector.Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on a body are all described by vectors.Many other physical quantities can be usefully thought of as vectors.One has to keep in mind, however, that the components of a physical vector depend on the coordinate system used to describe it.Use in physics and engineering
1.Vectors in Cartesian space
1.Euclidean vectors and affine vectors
1.Representation of a vector
3 Addition and scalar multiplication
3.Vector addition and subtraction
3.Length and the dot product
4.Length of a vector
4.Overview
Informally, a vector is a quantity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction, often represented graphically by an arrow.Sometimes, one speaks of bound or fixed vectors, which are vectors whose initial point is the origin.This is in contrast to free vectors, which are vectors whose initial point is not necessarily the origin.Use in physics and engineering
Vectors are fundamental in the physical sciences.For example, the velocity 5 meters per second upward could be represented by the vector (0,5).Another quantity represented by a vector is force, since it has a magnitude and direction.Vectors also describe many other physical quantities, such as displacement, acceleration, electric and magnetic fields, momentum, and angular momentum.Vectors in Cartesian space
In Cartesian coordinates, a vector can be represented by identifying the coordinates of its initial and terminal point.Typically in Cartesian coordinates, one considers primarily bound vectors.The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion.Euclidean vectors and affine vectors
In the geometrical and physical settings, sometimes it is possible to associate, in a natural way, a length to vectors as well as the notion of an angle between two vectors.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 spaces (for bound vectors) and affine spaces (for free vectors).Generalizations
In more general sorts of coordinate systems, rotations of a vector (and also of tensors) can be generalized and categorized to admit an analogous characterization by their covariance and contravariance under changes of coordinates.In mathematics, a vector is considered more than a representation of a physical quantity.In general, a vector is any element of a vector space over some field.The spatial vectors of this article are a very special case of this general definition (they are not simply any element of Rd in d dimensions), which includes a variety of mathematical objects (algebras, the set of all functions from a given domain to a given linear range, and linear transformations).Note that under this definition, a tensor is a special vector.Representation of a vector
Vectors are usually denoted in boldface, as a.Here the point A is called the initial point, tail, or base; point B is called the head, tip, or endpoint.The length of the arrow represents the vector's magnitude, while the direction in which the arrow points represents the vector's direction.These vectors are commonly shown as small circles.These can be thought of as viewing the tip an arrow front on and viewing the vanes of an arrow from the back.Cartesian plane, with endpoint (2,3).The vector itself is identified with its endpoint.In order to calculate with vectors, the graphical representation may be too cumbersome.The endpoint of a vector can be identified with a list of n real numbers, sometimes called a row vector or column vector.In three dimensional Euclidean space (or R3), vectors are identified with triples of numbers corresponding to the Cartesian coordinates of the endpoint (a,b,c).These have the intuitive interpretation as vectors of unit length pointing up the x, y, and z axis, respectively.Note: In introductory physics classes, these three special vectors are often instead denoted i, j, k (or when in Cartesian coordinates), but such notation clashes with the index notation and the summation convention commonly used in higher level mathematics, physics, and engineering.The use of Cartesian unit vectors as a basis in which to represent a vector, is not mandated.Vectors can also be expressed in terms of cylindrical unit vectors or spherical unit vectors .The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry respectively.Vector equality
Two vectors are said to be equal if they have the same magnitude and direction.However if we are talking about free vectors, then two free vectors are equal if they have the same base point and end point.For example, the vector e1 + 2e2 + 3e3 with base point (1,0,0) and the vector e1+2e2+3e3 with base point (0,1,0) are different free vectors, but the same (displacement) vector.Note: they only need to be linearly independent, i.The addition may be represented graphically by placing the start of the arrow b at the tip of the arrow a, and then drawing an arrow from the start of a to the tip of b.If a and b are free vectors, then the addition is only defined if a and b have the same base point, which will then also be the base point of a + b.Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the ends of a and b at the same point, and then draw an arrow from the tip of b to the tip of a.If a and b are free vectors, then the subtraction is only defined if they share the same base point which will then also become the base point of their difference.In the context of spatial vectors, 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.Scalar multiplication of a vector by a factor of 3 stretches the vector out.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.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.Pythagorean theorem since the basis vectors e1 , e2 , e3 are orthogonal unit vectors.Vector length and units
If a vector is itself spatial, the length of the arrow depends on a dimensionless scale.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.If you have a vector of arbitrary length, you can divide it by its length to create a unit vector.This is known as normalizing a 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 , or 0, or simply 0.Unlike any other vector, it does not have a direction, and cannot be normalized (i.The sum of the null vector with any vector a is a (i.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 cross product (also called the vector product or outer product) differs from the dot product primarily in that the result of the cross product of two vectors is a vector.While everything that was said above can be generalized in a straightforward manner to more than three dimensions, the cross product is only meaningful in three dimensions, although the seven dimensional cross product is similar in some respects.The problem with this definition is that there are two unit vectors perpendicular to both b and a.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.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 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.Illustration of tangential and normal components of a vector to a surface.When used in this role, the choice of their constituting directions is dependent upon the particular coordinate system being used, such as Cartesian coordinates, spherical coordinates or polar coordinates.For example, axial component of a vector is such that its component whose direction is determined by one of the Cartesian coordinate axes, whereas radial and tangential components relate to the radius of rotation of an object as their direction of reference.Both remain orthogonal to the axis of rotation at all times.In two dimensions this requirement becomes redundant as the axis degenerates to a point of rotation.The choice of a coordinate system doesn't affect properties of a vector or its behaviour under transformations.We can rewrite the directional derivative in differential form (without a given function f) as
Therefore any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative.Vectors, pseudovectors, and transformations
An alternative characterization of spatial vectors, especially in physics, describes vectors as lists of quantities which behave a certain way under a coordinate transformation.In other words, if all of space were rotated, the vector would rotate in exactly the same way.This important requirement is what distinguishes a spatial vector from any other triplet of physically meaningful quantities.On the other hand, for instance, a triplet consisting of the length, width, and height of a rectangular box could be regarded as the three components of an abstract vector, but not a spatial vector, since rotating the box does not correspondingly transform these three components.Examples of vectors include 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 vector to be a tensor of contravariant rank one.However, in differential geometry and other areas of mathematics such as representation theory, the "coordinate transitions" need not be restricted to rotations.Other notions of spatial vector correspond to different choices of symmetry group.As a particular case where the symmetry group is important, all of the above examples are vectors which "transform like the coordinates" under both proper and improper rotations.An example of an improper rotation is a mirror reflection.That is, these vectors are defined in such a way that, if all of space were flipped around through a mirror (or otherwise subjected to an improper rotation), that vector would flip around in exactly the same way.Vectors with this property are called true vectors, or polar vectors.However, other vectors are defined in such a way that, upon flipping through a mirror, the vector flips in the same way, but also acquires a negative sign.These are called pseudovectors (or axial vectors), and most commonly occur as cross products of true vectors.One example of an axial vector is angular momentum.Driving in a car, and looking forward, each of the wheels has an angular momentum 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 momentum vector points to the right, but the actual angular momentum 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.For historical development of the word vector, see "vector n.".Earliest Known Uses of Some of the Words of Mathematics.Mathematical treatments of spatial vectors
Apostol, T.Variable Calculus with an Introduction to Linear Algebra.Vectors, Tensors and the Basic Equations of Fluid Mechanics."Chapter 11", The Feynman Lectures on Physics, Volume I, 2nd ed, Addison Wesley.This page was last modified on 9 May 2008, at 13:00.All text is available under the terms of the GNU Free Documentation License.WHAT ARE YOU DOING THIS SUMMER?Get off to a great start with Vector.How will you know for sure?At Vector, our student work program builds entrepreneurs.Apply now and join the thousands of students who are getting off to a great start with Vector."He said there were three reasons he hired me..."This is a kid who in high school wouldn't get on the phone to order pizza.Go out and meet the people at Vector.DNA segment into a host cell.To learn more about vector visit Britannica.An element of a vector space.To guide (a pilot or aircraft, for example) by means of radio communication according to vectors.Published by Houghton Mifflin Company.An organism, such as a mosquito or tick, that spreads pathogens from one host to another.Published by Houghton Mifflin Company.Share This
vector In physics and mathematics, any quantity with both a magnitude and a direction.For example, velocity is a vector because it describes both how fast something is moving and in what direction it is moving.Because velocity is a vector, other quantities in which velocity is a factor, such as acceleration and momentum, are vectors also.Published by Houghton Mifflin Company.Published by Houghton Mifflin Company.By changing the vector to point to a different piece of code it is possible to modify the behaviour of the operating system.Vectors are said to be equal when their directions are the same their magnitudes equal.Note: In a triangle, either side is the vector sum of the other two sides taken in proper order; the process finding the vector sum of two or more vectors is vector addition (see under Addition).Web Search powered by Google
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