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Vector (spatial)

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  • sheilamarieflores_03
    Vector (spatial) From Wikipedia, the free encyclopedia (Redirected from Vector sum) Jump to: navigation, search This article is about vectors that have a
    Message 1 of 18 , Dec 1, 2007
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      Vector (spatial)
      From Wikipedia, the free encyclopedia
      (Redirected from Vector sum)
      Jump to: navigation, search
      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.

      A vector going from A to B.A spatial vector, or simply vector, is a
      concept characterized by a magnitude and a direction, and which sums
      with other vectors according to the Parallelogram law. 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


      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. [1]

      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.




      Contents [hide]
      1 Overview
      1.1 Use in physics and engineering
      1.2 Vectors in Cartesian space
      1.3 Euclidean vectors and affine vectors
      1.4 Generalizations
      2 Representation of a vector
      3 Addition and scalar multiplication
      3.1 Vector equality
      3.2 Vector addition and subtraction
      3.3 Scalar multiplication
      4 Length and the dot product
      4.1 Length of a vector
      4.1.1 Vector length and units
      4.2 Unit vector
      4.3 Dot product
      5 Cross product
      5.1 Scalar triple product
      6 Vector components
      7 Vectors as directional derivatives
      8 Pseudovectors
      9 References
      10 See also
      11 External links



      [edit] 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.


      [edit] 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 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 and magnetic fields,
      momentum, and angular momentum.


      [edit] 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 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



      [edit] 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 spaces (for bound vectors) and
      affine spaces (for free vectors).


      [edit] 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.


      [edit] 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 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.


      A vector in the 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. 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


      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:



      Another way to express a vector in three dimensions is to introduce
      the three basic coordinate vectors, sometimes referred to as unit
      vectors:


      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:


      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. This article will choose to use e1, e2, e3.

      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.


      [edit] Addition and scalar multiplication

      [edit] 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.


      [edit] Vector addition and subtraction
      Let a=a1e1 + a2e2 + a3e3 and b=b1e1 + b2e2 + b3e3, 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:


      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:


      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.





      [edit] 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:



      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.

      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 distributive over vector addition in the
      following sense: r(a + b) = ra + rb 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


      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/s² : 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.


      [edit] Length and the dot product

      [edit] Length of a vector
      The length or magnitude or norm of the vector a is denoted by ||a||
      or, less commonly, |a|, which is not to be confused with the
      absolute value (a scalar "norm").

      The length of the vector a = a1e1 + a2e2+ a3e3 in a three-
      dimensional Euclidean space, where e1, e2, e3 are orthogonal unit
      vectors, can be computed with the Euclidean norm


      which is a consequence of the Pythagorean theorem since the basis
      vectors e1 , e2 , e3 are orthogonal unit vectors.

      This happens to be equal to the square root of the dot product of
      the vector with itself:



      [edit] Vector length and units
      If a vector is itself spatial, the length of the arrow depends on a
      dimensionless scale. If it represents e.g. a force, the "scale" is
      of physical dimension length/force. Thus there is typically
      consistency in scale among quantities of the same dimension, but
      otherwise scale ratios may vary; for example, if "1 newton" and "5
      m" are both represented with an arrow of 2 cm, the scales are 1:250
      and 1 m:50 N respectively. Equal length of vectors of different
      dimension has no particular significance unless there is some
      proportionality constant inherent in the system that the diagram
      represents. Also length of a unit vector (of dimension length, not
      length/force, etc.) has no coordinate-system-invariant significance.


      [edit] Unit vector
      Main article: Unit vector a.k.a. Direction vector
      A unit vector is any vector with a length of one; geometrically, it
      indicates a direction but no magnitude. 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. A unit vector is
      often indicated with a hat as in â.


      To normalize a vector a = [a1, a2, a3], scale the vector by the
      reciprocal of its length ||a||. That is:



      [edit] Dot product
      Main article: Dot product
      The dot product of two vectors a and b (sometimes called the inner
      product, or, since its result is a scalar, the scalar product) is
      denoted by a ∙ b and is defined as:


      where ||a|| and ||b|| denote the 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.

      The dot product can also be defined as the sum of the products of
      the components of each vector:


      where a and b are vectors of n dimensions; a1, a2, …, an are
      coordinates of a; and b1, b2, …, bn are coordinates of b.

      This operation is often useful in physics; for instance, work is the
      dot product of force and displacement.


      [edit] Cross product
      Main article: Cross product
      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 cross product, denoted a × b, is a vector
      perpendicular to both a and b and is defined as:


      where θ is the measure of the angle between a and b, and n is a unit
      vector perpendicular to both a and b. The problem with this
      definition is that there are two unit vectors perpendicular to both
      b and a.


      An illustration of the cross product.The vector basis e1, e2 , e3 is
      called right handed, if the three vectors are situated like the
      thumb, index finger and middle finger (pointing straight up from
      your palm) of your right hand. Graphically the cross product can be
      represented by the figure on the right.

      The cross product a × b is defined so that a, b, and a × b also
      becomes a right handed system (but note that a and b are not
      necessarily orthogonal). This is the right-hand rule.

      The length of a × b can be interpreted as the area of the
      parallelogram having a and b as sides.

      The cross product of two vectors is a pseudovector (see below).


      [edit] Scalar triple product
      The scalar triple product (also called the box product or mixed
      triple product) is not really a new operator, but a way of applying
      the other two multiplication operators to three vectors. The scalar
      triple product is sometimes denoted by (a b c) and defined as:


      It has three primary uses. First, the absolute value of the box
      product is the volume of the parallelepiped which has edges that are
      defined by the three vectors. Second, the scalar triple product is
      zero if and only if the three vectors are linearly dependent, which
      can be easily proved by considering that in order for the three
      vectors to not make a volume, they must all lie in the same plane.
      Third, the box product is positive if and only if the three vectors
      a, b and c are right-handed.

      In components ( with respect to a right-handed orthonormal basis),
      if the three vectors are thought of as rows (or columns, but in the
      same order), the scalar triple product is simply the determinant of
      the 3-by-3 matrix having the three vectors as rows. The scalar
      triple product is linear in all three entries and anti-symmetric in
      the following sense:








      [edit] Vector components

      Illustration of tangential and normal components of a vector to a
      surface.A component of a vector is the influence of that vector in a
      given direction. [1] Components are themselves vectors.

      A vector is often described by a fixed number of components that sum
      up into this vector uniquely and totally. 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 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. The former is
      parallel to the radius and the latter is orthogonal to it. [2] 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.


      [edit] Vectors as directional derivatives
      A vector may also be defined as a directional derivative: consider a
      function f(xα) and a curve xα(σ). Then the directional derivative of
      f is a scalar defined as


      where the index α is summed over the appropriate number of
      dimensions (e.g. from 1 to 3 in 3-dimensional Euclidian space, from
      0 to 3 in 4-dimensional spacetime, etc.). Then consider a vector
      tangent to xα(σ):


      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. We can therefore define a
      vector precisely:



      [edit] Pseudovectors
      Main article: pseudovector
      Many 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. These are called true vectors, or polar vectors, and
      include such things as a particle's position and velocity (in
      physics), or a vector defined as a directional derivative (see
      above). 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.
    • jaimemanangan
      A spatial vector, or simply vector, is a geometric object that has both a magnitude and a direction. A vector is frequently represented by a line segment
      Message 2 of 18 , Jul 3, 2008
      • 0 Attachment
        A spatial vector, or simply vector, is a geometric object that has
        both a magnitude and a direction. A vector is frequently represented
        by a line segment connecting the initial point A with the terminal
        point B and denoted


        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.[1]

        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. The sum of two vectors with the same
        initial point can be found geometrically using the parallelogram
        law. Multiplication by a positive number, commonly called a scalar
        in this context, amounts to changing the magnitude of vector, that
        is, stretching or compressing it while keeping its direction;
        multiplication by -1 preserves the magnitude of the vector but
        reverses its direction.

        Cartesian coordinates provide a systematic way of describing vectors
        and operations on them. A vector becomes a tuple of real numbers,
        its scalar components. 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. Other vector-like objects that describe physical quantities and
        transform in a similar way under changes of the coordinate system
        include pseudovectors and tensors.
      • stephanebacolcol_ivgold
        A spatial vector, or simply vector, is a geometric object that has both a magnitude and a direction. A vector is frequently represented by a line segment
        Message 3 of 18 , Jul 6, 2008
        • 0 Attachment
          A spatial vector, or simply vector, is a geometric object that has
          both a magnitude and a direction. A vector is frequently represented
          by a line segment connecting the initial point A with the terminal
          point B and denoted


          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.[1]

          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. The sum of two vectors with the same
          initial point can be found geometrically using the parallelogram law.
          Multiplication by a positive number, commonly called a scalar in this
          context, amounts to changing the magnitude of vector, that is,
          stretching or compressing it while keeping its direction;
          multiplication by -1 preserves the magnitude of the vector but
          reverses its direction.

          Cartesian coordinates provide a systematic way of describing vectors
          and operations on them. A vector becomes a tuple of real numbers, its
          scalar components. 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.
          Other vector-like objects that describe physical quantities and
          transform in a similar way under changes of the coordinate system
          include pseudovectors and tensors.
        • Josiah Garcia
          A spatial vector, or simply vector, is a geometric object that has both a magnitude and a direction. A vector is frequently represented by a line segment
          Message 4 of 18 , Jul 19, 2008
          • 0 Attachment
            A spatial vector, or simply vector, is a geometric object that has
            both a magnitude and a direction. A vector is frequently represented
            by a line segment connecting the initial point A with the terminal
            point B and denoted
          • kentramos47
            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
            Message 5 of 18 , Aug 8, 2008
            • 0 Attachment
              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.
              Other vector-like objects that describe physical quantities and
              transform in a similar way under changes of the coordinate system
              include pseudovectors and tensors.
            • kentramos47
              A spatial vector, or simply vector, is a geometric object that has both a magnitude and a direction. A vector is frequently represented by a line segment
              Message 6 of 18 , Aug 8, 2008
              • 0 Attachment
                A spatial vector, or simply vector, is a geometric object that has
                both a magnitude and a direction. A vector is frequently represented
                by a line segment connecting the initial point A with the terminal
                point B and denoted


                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.[1]

                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. The sum of two vectors with the same
                initial point can be found geometrically using the parallelogram law.
                Multiplication by a positive number, commonly called a scalar in this
                context, amounts to changing the magnitude of vector, that is,
                stretching or compressing it while keeping its direction;
                multiplication by -1 preserves the magnitude of the vector but
                reverses its direction.

                Cartesian coordinates provide a systematic way of describing vectors
                and operations on them. A vector becomes a tuple of real numbers, its
                scalar components. Addition of vectors and multiplication of a vector
                by a scalar are simply done component by component, see coordinate
                vector.
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