Vector Mathematics

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Vector Mathematics

• Adding, Subtracting, Multiplying and Dividing

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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

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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

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Math Coordinate System (Direction)
90º
180º
0º
270º
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Polar Coordinate System (Direction and Magnitude)
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Polar ?? Math (Cartesian)
r
y
?
x
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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

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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
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Important

• You can add vectors in any order and yield the
  same resultant.

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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

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Components of Vectors

• A Ax Ay
• Ax A cos ?
• Ay A sin ?
• As long as you draw the x component first

A
Ay
?
Ax
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The Table Method

• We will organize these components in a table.
• See the board for this part and next slide

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Table Method Equation

• Add all X components together ? Final Rx
• Add all Y components together ? Final Ry

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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)

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Subtracting Vectors
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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)

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Unit Vectors for vectors A B
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Unit Vectors
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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

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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

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Multiplying Vectors (products)

• 2. (vector) (vector) Scalar
• This is called the Scalar Product or the Dot
  Product

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Dot Product Continued (see p. 25)
B
F
A
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Multiplying Vectors (products)

• (vector) x (vector) vector
• This is called the vector product or the cross
  product

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Cross Product Continued
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Cross Product Direction and reverse
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Cross Product

• You can also solve the Cross Product with a
  matrix and unit vectorscheck out the board for
  this.