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Vectors

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					                      Vectors
• Recall that “vectors” are arrows that
  represent a vector quantity (magnitude and
  direction).

  – The length of the arrow represents the
    magnitude of a measurement
  – The direction of the arrow within a coordinate
    system represents the direction of the vector.
     • Most vector directions are references to a particular
       direction, i.e., 0o or say the “west” or negative x-axis
       of a graph.
               Vectors move!
• Vector quantities of the same units can be added
  to each other.
• Vectors can be added in 2 ways:
   – Geometrically
   – Algebraically
• To add 2 vectors geometrically simply place the
  tail of one vector to the head of the other. The
  solution, or Resultant vector, is the vector from the
  point of origin of the 1st vector to the end point of
  the 2nd.
• When adding vectors geometrically, one often
  places the first vector at the origin.
              Example:
                                     Fb
Fa       Fb
                         R           Fa


         Fb


     R                       R
         Fa
                                 

                                 R = magnitude R
                                    @ o North
                                     of East
  Adding Vectors Algebraically
• Adding vectors algebraically involves
  simply adding magnitudes that are along the
  same axes.

             The Unit Vector
• A unit vector is a vector of 1 “unit” in
  length that defines a particular direction.
   – For example: In a Cartesian system, there are 3
     principal axes: x, y, & z. The unit vector of
     each is simply a vector of length 1, in each
     direction.
    • Often a vector may be written as the sum of
      its parts, each multiplied by a unit vector
      giving the direction associated with that
      part.
y
                    The vector R shown at the right
     R              could be written as R = 3x + 4y
          4

              x     Notice that the vector R can be
      3             Represented as the geometric sum of
                    3 times the x-unit vector plus 4 times
                    The y-unit vector.
                     Notation


• i, j, k notation



• Arrows



• Boldface
            Algebraic Sums
• The resultant vector obtained from the
  graphical addition of two vectors can be
  found form adding the vectors algebraically.
1st: Break each vector into independent
  components. I.e., Put the vector into i,j,k
  notation if it’s not already.
2nd: Add each component of both vectors
  independently of the other components.
• Example
  – A = 3i + 4j +5k
  – B = 2i +7k
  R=A+B
  R = 3i + 4j +5k + 2i +7k   OR     3i + 4j +5k
                                  + 2i + 0j +7k
  R = 5i + 4j + 12k
     Resolution of Vectors into
           Components
• Vectors describing real conditions are rarely
  written in the easy-to-use Cartesian
  notation. More often a vector is expressed
  as an angle.
• For example, the velocity of a projectile
  might be given as 30 m/s at 25o above the
  plain.
• In order to work with this velocity vector in
  a meaningful context we must often
  “resolve” it, or break it down into its
  component parts.

• Generally we will employ the use of
  trigonometry to accomplish this task.
                            V = Vx + Vy
                            Vx  Vcos
                            Vy  Vsin
                  Angles
• To find the angle formed when two
  independent (i.e., x- and y- dimension)
  vectors are added simply use the ratio of
  their magnitudes and the tangent function.
            Known Vx & Vy
                      Vy
            tan  
                      Vx
                       Vy 
              tan 
                   1
                           
                       Vx 
                          
         Independent motion
• Since we are using vectors as variables to
  analyze motion it is critical to note that
  vectors that are at right angles to each other
  are independent.
• This means that motion in any one direction
  does not affect motion at right angles to that
  referenced direction.
  – For example: Up/Down (Vertical) motion is
    independent of Side/Side (Horizontal) motion
    during projectile flight.
            Products of Vectors

• The Scalar Product
  – The “dot” product
 The scalar product of two vectors is the product of the
 magnitude of one vector and the component of a second
 vector along the direction of the first.

 The scalar or dot product takes the form:

 a  b = a b cos 
A  B  A BA  A B cos
         The Vector Product
          (Cross Product)
• The cross product of two vectors is given by

           R  A B sin 
where  is the smallest angle between the two
  vectors.
• The direction of the resultant vector is
  perpendicular to the plane defined by
  vectors A & B and given by the right hand
  rule.
    RHR Practice
In what direction is R?

				
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