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Vectors
Scalars and Vectors
• A scalar is a single number that represents a
  magnitude
   – E.g. distance, mass, speed, temperature, etc.

• A vector is a set of numbers that describe both a
  magnitude and direction
   – E.g. velocity (the magnitude of velocity is
     speed), force, momentum, etc.

• Notation: a vector-valued variable is
  differentiated from a scalar one by using bold or
  the following symbol:
                         
           A             a                 2
Characteristics of Vectors
A Vector is something that has two and
   only two defining characteristics:

1. Magnitude: the 'size' or 'quantity'

2. Direction: the vector is directed from
   one place to another.

                                3
Direction
• Speed vs. Velocity
• Speed is a scalar, (magnitude no direction) -
  such as 5 feet per second.
• Speed does not tell the direction the object
  is moving. All that we know from the speed is
  the magnitude of the movement.
• Velocity, is a vector (both magnitude and
  direction) – such as 5 ft/s Eastward. It tells
  you the magnitude of the movement, 5 ft/s,
  as well as the direction which is Eastward.

                                     4
Example

•The direction of the vector is
55° North of East

•The magnitude of the vector
is 2.3.




                    5
Now You Try


Direction: 47° North of West

Magnitude: 2




                   6
Try Again




Direction: 43° East of South

Magnitude: 3

                        7
Try Again




It is also possible to describe this
vector's direction as 47 South of East.

              Why?
                             8
Expressing Vectors as Ordered
            Pairs

        How can we express this
         vector as an ordered pair?

             Use Trigonometry




                          9
10
Now You Try




Express this vector as an ordered pair.



                                11
Adding Vectors
    Add vectors A and B

y




     A
         B
                   x


                       12
Adding Vectors
On a graph, add vectors using the “head-to-tail” rule:


                y




                    A
                            B
                                                   x
     Move B so that the head of A touches the tail of B
Note: “moving” B does not change it. A vector is only defined by its
magnitude and direction, not starting location.            13
Adding Vectors
The vector starting at the tail of A and ending at the
  head of B is C, the sum (or resultant) of A and B.

           y
                       B
                                     C = A+ B
               A
                       C
                                     x


                                           14
Adding Vectors
• Note: moving a
  vector does not
  change it. A vector
  is only defined by
  its magnitude and
  direction, not
  starting location




                              15
Adding Vectors
Let’s go back to our example:
   y


           1,5


       A
                 B               7,1

                                                    x
                     Now our vectors have values.




                                                        16
Adding Vectors
What is the value of our resultant?

         y
                         7,1
                    B
             1,5


              A
                    C
                               x

GeoGebra Investigation
                                   17

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Vectors

  • 2. Scalars and Vectors • A scalar is a single number that represents a magnitude – E.g. distance, mass, speed, temperature, etc. • A vector is a set of numbers that describe both a magnitude and direction – E.g. velocity (the magnitude of velocity is speed), force, momentum, etc. • Notation: a vector-valued variable is differentiated from a scalar one by using bold or the following symbol:  A a 2
  • 3. Characteristics of Vectors A Vector is something that has two and only two defining characteristics: 1. Magnitude: the 'size' or 'quantity' 2. Direction: the vector is directed from one place to another. 3
  • 4. Direction • Speed vs. Velocity • Speed is a scalar, (magnitude no direction) - such as 5 feet per second. • Speed does not tell the direction the object is moving. All that we know from the speed is the magnitude of the movement. • Velocity, is a vector (both magnitude and direction) – such as 5 ft/s Eastward. It tells you the magnitude of the movement, 5 ft/s, as well as the direction which is Eastward. 4
  • 5. Example •The direction of the vector is 55° North of East •The magnitude of the vector is 2.3. 5
  • 6. Now You Try Direction: 47° North of West Magnitude: 2 6
  • 7. Try Again Direction: 43° East of South Magnitude: 3 7
  • 8. Try Again It is also possible to describe this vector's direction as 47 South of East. Why? 8
  • 9. Expressing Vectors as Ordered Pairs How can we express this vector as an ordered pair? Use Trigonometry 9
  • 10. 10
  • 11. Now You Try Express this vector as an ordered pair. 11
  • 12. Adding Vectors Add vectors A and B y A B x 12
  • 13. Adding Vectors On a graph, add vectors using the “head-to-tail” rule: y A B x Move B so that the head of A touches the tail of B Note: “moving” B does not change it. A vector is only defined by its magnitude and direction, not starting location. 13
  • 14. Adding Vectors The vector starting at the tail of A and ending at the head of B is C, the sum (or resultant) of A and B. y B C = A+ B A C x 14
  • 15. Adding Vectors • Note: moving a vector does not change it. A vector is only defined by its magnitude and direction, not starting location 15
  • 16. Adding Vectors Let’s go back to our example: y 1,5 A B 7,1 x Now our vectors have values. 16
  • 17. Adding Vectors What is the value of our resultant? y 7,1 B 1,5 A C x GeoGebra Investigation 17