Vectors by 7ROYwz6

VIEWS: 14 PAGES: 17

									Vectors
            Scalars & Vectors
• Vectors
  – Quantity with both
    magnitude & direction
  – Does NOT follow                                    Head
    elementary
    arithmetic/algebra rules
                                            Direction/Angle
  – Examples – position,       Tail

    force, moment,             Line of Action
    velocities, acceleration
           Parallelogram Law
• The resultant of two forces
  can be obtained by
   – Joining the vectors at their
     tails
                                      A
                                              A+B
      Constructing a
                                          B
       parallelogram
      The resultant is the diagonal of the parallelogram
          Triangle Construction
• The resultant of two
  forces can be obtained
  by
   – Joining the vectors in
     tip-to-tail fashion         A       B
                                     R
        The resultant extends
         from the tail of A to
         the head of the B
         Vector Addition
• Does
         A+B       =    B+A           ?



     A         B
          R                       R
                              B           A


              YES! - commutative
Vector Subtraction
A-B               =   A + (-B)


A                        -B
          B




          -B
      R
              A
             Vector Subtraction
• Does
             A–B      =    B-A             ?



               -B                     B
         R
               A              -R      -A

               NO! – opposite sense
           Vector Operations
• Multiplication & Division of Vector (A) by
  Scalar (a)
     a * A = aA


                                   2A
       2 * A = 2A        A


      -.5 * A = -.5A     A        -.5A
      Representation of a Vector
   Given the points A( x1 , y1 , z1 ) and B( x2 , y2 , z2 ),
   the vector a with representation AB is
                 a  x2  x1, y2  y1, z2  z1

Find the vector represented by the directed line segment with
      initial point A(2,-3,4) and terminal point B(-2,1,1).

                  a  2  2,1  (3),1  4

                     a  4, 4, 3
    Magnitude of a vector


Determine the magnitude of the following:
                                  Example
If a  4,0,3 and b  2,1,5 , find a and the vectors a+b, a-b, 3b,and 2a+5b.

                                           a  b  4, 0,3  2,1,5
  a  4  0  3  25  5
        2    2   2

                                           a  b  4  2, 0  1,3  5
                                           a  b  2,1,8

                                   3b  3 2,1,5  3(2),3(1),3(5)  6,3,15
  a  b  4, 0,3  2,1,5
  a  b  4  (2), 0  1,3  5
                                     2a  5b  2 4, 0,3  5 2,1,5
  a  b  6, 1, 2
                                     2a  5b  8, 0, 6  10,5, 25
                                     2a  5b  2,5,31
                                     Parallel
•   Two vectors are parallel to each other if one is the scalar multiple of the other.

Determine if the two vectors are parallel




      These are parallel since                        These are not parallel
      b= -3a                                          since 4(1/2) =2 , but
                                                      10(1/2)=5 not -9
                       Unit vectors
  Any vector that has a magnitude of 1 is considered a unit vector.


  Can you think of a unit vector?



i= 1,0,0                     j= 0,1,0                       k= 0,0,1
          Standard Basis Vectors
If a= a1 , a2 , a3 , then we can write
a= a1 , a2 , a3  a1 , 0, 0  0, a2 , 0  0, 0, a3

a=a1 1,0,0  a2 0,1,0  a3 0,0,1
a=a1i  a2 j  a3k
Example- Write 1, 2,6 in terms of the standard basis vector i,j,k.



             1, 2,6  i - 2j  6k
                 Example
If a = i + 2j - 3k and b = 4i + 7k, express the
   vector 2a+3b in terms of i,j,k.

2a+3b=2(i + 2j - 3k)+3(4i + 7k)
2a+3b=2i + 4j - 6k+ 12i + 21k
2a+3b=14i+4j+15k
                           Unit Vectors
                                                         1    a
    The unit vector in the same direction of a is   u=     a=
                                                         a    a
    Find a unit vector in the same direction as 2i – j – 2k.
We are looking for a vector in the same direction as the original vector, but is also
a unit vector.
Let’s first find the magnitude   2i - j - 2k  22  (1)2  (2)2  9  3
    1   a
  u= a=                          Check?
    a   a
                                 Same direction?
    1
  u= a                           Magnitude = 1?
    3
    1
  u= (2i - j - 2k)
    3
    2 1         2
  u= i - j - k
    3 3         3
               Homework
• P649
  – 4,5,7,9,11,15,17,19

								
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