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Vector Mathematics

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... and the resultant is the straight line from start to finish ... Check out the next s... Components of Vectors. A = Ax Ay. Ax =A cos ?. Ay = A sin ? ... – PowerPoint PPT presentation

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Title: Vector Mathematics


1
Vector Mathematics
  • Adding, Subtracting, Multiplying and Dividing

2
Why?
  • One can add 23 kg and 42 kg and get 65 kg.
  • However, one cannot add together 23 m/s south and
    42 m/s southeast and get 65 m/s south-southeast.
  • Vectors addition takes into account adding both
    magnitude and direction

3
Words
  • Vector A measured quantity with both magnitude
    (the how big part) and direction
  • Scalar A measured quantity with magnitude only
  • Resultant Vector The final vector of a vector
    math problem

4
Math Coordinate System (Direction)
90º
180º

270º
5
Polar Coordinate System (Direction and Magnitude)
6
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7
Polar ?? Math (Cartesian)
r
y
?
x
8
Vector addition
  • Two Ways
  • Graphically Draw vectors to scale, Tip to Tail,
    and the resultant is the straight line from start
    to finish
  • Mathematically Employ vector math analysis to
    solve for the resultant vector

9
Graphically 2-D Right
  • A 5.0 m _at_ 0
  • B 5.0 m _at_ 90
  • Solve A B

R
R7.1 m _at_ 45
Start
10
Important
  • You can add vectors in any order and yield the
    same resultant.

11
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12
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13
Lets add the last one mathematically
  • The math you used previously doesnt work (and I
    wont let you use the Law of Sines or Cosines) or
    does it???
  • What we will do is break each vector into
    components
  • The components are the x and y values of the
    polar coordinate (go back 6 slides)
  • Check out the next slides

14
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15
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16
Components of Vectors
  • A Ax Ay
  • Ax A cos ?
  • Ay A sin ?
  • As long as you draw the x component first

A
Ay
?
Ax
17
The Table Method
  • We will organize these components in a table.
  • See the board for this part and next slide

18
Table Method Equation
  • Add all X components together ? Final Rx
  • Add all Y components together ? Final Ry

19
Subtracting Vectors
  • Simply add or subtract 180 (keep ? between 0
    and 360) to the direction of the vector being
    subtracted
  • You just ADD the OPPOSITE vector (there is no
    subtraction in vector math)

20
Subtracting Vectors
21
Unit Vectors
  • A unit vector is a vector that has a magnitude of
    1, with no units.
  • Its only purpose is to point
  • We will use i, j, k for our unit vectors
  • i means x direction, j is y, and k is z
  • We also put little hats () on i, j, k to show
    that they are unit vectors (I will boldface them)

22
Unit Vectors for vectors A B
23
Unit Vectors
24
Adding using unit vectors
  • R A B
  • R (Ax Bx )i (Ay By )j (Az Bz )kwhich
    becomes R Rx i Ry j Rz k
  • The magnitude of R is found by applying the
    Pythagorean theorem

25
Multiplying Vectors (products)3 ways
  • Scalar x Vector Vector w/ magnitude multiplied
    by the value of scalar A 5 m _at_ 303A 15m
    _at_ 30

26
Multiplying Vectors (products)
  • 2. (vector) (vector) Scalar
  • This is called the Scalar Product or the Dot
    Product

27
Dot Product Continued (see p. 25)
B
F
A
28
Multiplying Vectors (products)
  • (vector) x (vector) vector
  • This is called the vector product or the cross
    product

29
Cross Product Continued
30
Cross Product Direction and reverse
31
Cross Product
  • You can also solve the Cross Product with a
    matrix and unit vectorscheck out the board for
    this.
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