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In Physics and in Vector Calculus , a spatial vector, or simply '''vector''', is a concept characterized by a Magnitude and a direction. A vector can be thought of as an arrow in Euclidean Space , drawn from an '''initial point''' ''A'' pointing to a '''terminal point''' ''B''. This vector is commonly denoted by :$\backslash overrightarrow\{AB\},$ indicating that the arrow points from ''A'' to ''B''. In this way, the arrow holds all the information of the vector quantity — the magnitude is represented by the arrow's length and the direction by the direction of the arrow's head and body. This magnitude and direction are those necessary to carry one from ''A'' to ''B''. Indeed in Latin the word ''vector'' means "one who carries"; Latin ''veho'' = "I carry". For historical development of the word ''vector'', see . See also 1 here the vector is what would carry a point from ''A'' to ''B''. Vectors have a variety of Algebraic properties. Vectors may be Scaled by stretching them out, or compressing them. They can be flipped around so as to point in the opposite direction. Two vectors sharing the same initial point can also be Added or Subtracted . 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 the '''free''' vectors, which are not necessarily attached to the origin. Use in physics and engineering Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has both a magnitude and direction, such as Velocity , the magnitude of which is Speed . For example, the velocity "''5 $frac\{meters\}\{second\}$ up''" 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 Field , 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. For instance, the points ''A'' = (1,0,0) and ''B'' = (0,1,0) in space determine the free vector $\backslash overrightarrow\{AB\}$ pointing from the point ''x''=1 on the ''x''-axis to the point ''y''=1 on the ''y''-axis. Typically in Cartesian coordinates, one considers primarily bound vectors. A bound vector is determined by the coordinates of the terminal point, its initial point always having the coordinates of the origin ''O'' = (0,0,0). Thus the bound vector represented by (1,0,0) is a vector of unit length pointing from the origin up the positive ''x''-axis. The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion. For example, the sum of the vectors (1,2,3) and (-2,0,4) is the vector :$(1,\backslash ,\; 2,\backslash ,\; 3)\; +\; (-2,\backslash ,\; 0,\backslash ,\; 4)=(1-2,\backslash ,\; 2+0,\backslash ,\; 3+4)=(-1,\backslash ,\; 2,\backslash ,\; 7).\backslash ,$ 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. 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 and angle. In three-dimensions, it is further possible to define a ''cross product'' which supplies an algebraic characterization of area. 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). Generalizations In more general sorts of coordinate systems, rotations of a vectors (and also of Tensor s) 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 ''' R '''''d'' in ''d'' dimensions), which includes a variety of mathematical objects ( Algebras , the Set of all Function s from a given Domain to a given linear Range , and Linear Transformation s). Note that under this definition, a Tensor is a special vector. REPRESENTATION OF A VECTOR Vectors are usually denoted in boldface, as a. Other conventions include or ''a'', especially in handwriting. Alternately, some use a Tilde (~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type. Vectors are usually shown in graphs or other diagrams as arrows, as illustrated below: Here the point ''A'' is called the ''tail'', ''base'', ''start'', or ''origin''; point ''B'' is called the ''head'', ''tip'', ''endpoint'', or ''destination''. The length of the arrow represents the vector's magnitude, while the direction in which the arrow points represents the vector's direction. In the figure above, the arrow can also be written as $\backslash overrightarrow\{AB\}$ or ''AB''. On a two-dimensional diagram, sometimes a vector Perpendicular to the Plane of the diagram is desired. These vectors are commonly shown as small circles. A circle with a dot at its centre indicates a vector pointing out of the front of the diagram, towards the viewer. A circle with a cross inscribed in it indicates a vector pointing into and behind the diagram. These can be thought of as viewing the tip an Arrow front on and viewing the vanes 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 in a Cartesian Coordinate System . The endpoint of a vector can be identified with a list of ''n'' real numbers, sometimes called a Row Vector or Column Vector . As an example in two dimensions (see image), the vector from the origin ''O'' = (0,0) to the point ''A'' = (2,3) is simply written as :$\backslash overrightarrow\{OA\}\; =\; (2,3).$ 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 numbers are often arranged into a column vector or row vector, particularly when dealing with Matrices , as follows: :$\backslash mathbf\{a\}\; =\; \backslash begin\{bmatrix\}$ a\ b\ c\ \end{bmatrix} :$\backslash mathbf\{a\}\; =\; \backslash langle\; a\backslash \; b\backslash \; c\; angle.$ Another way to express a vector in three dimensions is to introduce the three basic ''coordinate vectors'', sometimes referred to as unit vectors: :$\{\backslash mathbf\; e\}\_1\; =\; (1,0,0),\; \{\backslash mathbf\; e\}\_2\; =\; (0,1,0),\; \{\backslash mathbf\; e\}\_3\; =\; (0,0,1).$ These have the intuitive interpretation as vectors of unit length pointing up the ''x'', ''y'', and ''z'' axis, respectively. In terms of these, any vector in R3 can be expressed in the form: :$(a,b,c)\; =\; a(1,0,0)\; +\; b(0,1,0)\; +\; c(0,0,1)\; =\; a\{\backslash mathbf\; e\}\_1\; +\; b\{\backslash mathbf\; e\}\_2\; +\; c\{\backslash mathbf\; e\}\_3.$ Note: In introductory physics classes, these three special vectors are often instead denoted '''i''', '''j''', '''k''' (or $\backslash boldsymbol\{\backslash hat\{x\}\},\; \backslash boldsymbol\{\backslash hat\{y\}\},\; \backslash boldsymbol\{\backslash hat\{z\}\}$ 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. This article will choose to use '''e1''', '''e2''', '''e3'''. The use of Cartesian unit vectors $\backslash boldsymbol\{\backslash hat\{x\}\},\; \backslash boldsymbol\{\backslash hat\{y\}\},\; \backslash boldsymbol\{\backslash hat\{z\}\}$ as a Basis in which to represent a vector, is not mandated. Vectors can also be expressed in terms of Cylindrical unit vectors $\backslash boldsymbol\{\backslash hat\{r\}\},\; \backslash boldsymbol\{\backslash hat\{\; heta\}\},\; \backslash boldsymbol\{\backslash hat\{z\}\}$ or Spherical unit vectors $\backslash boldsymbol\{\backslash hat\{r\}\},\; \backslash boldsymbol\{\backslash hat\{\; heta\}\},\; \backslash boldsymbol\{\backslash hat\{\backslash phi\}\}$. The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry respectively. ADDITION AND SCALAR MULTIPLICATION 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 + 2'''e2''' + 3'''e3''' with base point (1,0,0) and the vector e1+2'''e2'''+3'''e3''' with base point (0,1,0) are different free vectors, but the same (displacement) vector. Vector addition and subtraction Let a=''a''1'''e1''' + ''a''2'''e2''' + ''a''3'''e3''' and '''b'''=''b''1'''e1''' + ''b''2'''e2''' + ''b''3'''e3''', where e1, '''e2''', '''e3''' are orthogonal unit vectors (Note: they only need to be linearly independent, i.e. not parallel and not in the same plane, for these algebraic addition and subtraction rules to apply) The sum of a and '''b''' is: :$\backslash mathbf\{a\}+\backslash mathbf\{b\}$ =(a_1+b_1)\mathbf{e_1} +(a_2+b_2)\mathbf{e_2} +(a_3+b_3)\mathbf{e_3} 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. The new arrow drawn represents the vector '''a''' + b, as illustrated below: 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 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'''. One can check geometrically that a + '''b''' = '''b''' + a and (a + '''b''') + '''c''' = a + ('''b''' + '''c'''). The difference of a and '''b''' is: :$\backslash mathbf\{a\}-\backslash mathbf\{b\}$ =(a_1-b_1)\mathbf{e_1} +(a_2-b_2)\mathbf{e_2} +(a_3-b_3)\mathbf{e_3} 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'''. That arrow represents the vector '''a''' − b, as illustrated below: 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. This operation deserves the name "subtraction" because (a − '''b''') + '''b''' = a. Scalar multiplication A vector may also be multiplied, or re-''scaled'', by a Real Number ''r''. 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'''. The resulting vector is: :$r\backslash mathbf\{a\}=(ra\_1)\backslash mathbf\{e\_1\}$ +(ra_2)\mathbf{e_2} +(ra_3)\mathbf{e_3} 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 compatible with 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. In physics, scalars may also have a unit of measurement associated with them. For instance, Newton's Second Law is :$\{\backslash mathbf\; F\}\; =\; m\{\backslash mathbf\; a\}$ 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&2 : 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. LENGTH AND THE DOT PRODUCT Length of a vector
	:<math>\left\\mathbf{a} Ight\	\sqrt{{a_1}^2+{a_2}^2+{a_3}^2}</math>
	To Normalize A Vector '''a'''	''a''<sub>2</sub>, ''a''<sub>3</sub> , scale the vector by the reciprocal of its length '''a''' That is:

  \left\\mathbf{a} ight\\left\\mathbf{b} ight\\cos heta</math>
  Where '''a''' And '''b''' Denote The "http://wwwinformationdelightinfo/information/entry/norm_(mathematics)" class="copylinks">Norm (or length) of '''a''' and '''b''', and ''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'''
  \left\\mathbf{a} ight\\left\\mathbf{b} ight\\sin( heta)\,\mathbf{n}</math>