Mathematics with vectors

1
Mathematics with vectors

• When dealing with more than one dimension, can
  have the same equation come up multiple times.
• Example Position equation in three dimensions
  yields

2

• In a situation with multiple objects and multiple
  physical quantities, the number of equations
  increases dramatically!
• Need a shorthand to simplify the process.
• Vectors provide this simplification.
• Note that vectors provide a shorthand to see
  physical relationships, but never allow for
  numerical answers!

3
Definition of vectors

• Vectors are a set of numbers that describe the
  same physical quantity
• Number of numbers equals the number of dimensions
• Example Blood pressure use both systolic and
  diastolic to determine health
• Vectors are physical objects with magnitude and
  direction
• Example Herd migration number of animals and
  direction of movement determines migration.

4
Components

• Magnitude of vector depicted by length
• Direction by angle
• Components are projection onto the various axes
• Length along axis is magnitude in that direction
• One component for each axis

5
Shorthand Notation

• is shorthand for

where letter subscripts indicate projection axis
6
Addition/Subtraction

• Add and subtract vectors by components
• Notice that if B has a component in an opposite
  direction of A, then components subtract.

• Subtract vectors by reversing direction of
  subtracted vector and adding
• Reverse direction by changing sign on all
  components.

7
Vector Multiplication

• Three types of vector multiplication
• scalar x vector vector
• vector x vector scalar
• vector x vector vector
• Scalar (number) x vector simply stretches
  vector
• Scalar multiplies each component equally

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Scalar (Dot) Product

• vector x vector scalar
• Product of magnitude of one vector times
  projected magnitude of other

9
Vector (Cross) Product

• vector x vector vector
• Two (perpendicular) vectors define plane
• Direction of product is at right angles to this
  plane
• Magnitude is product of perpendicular vectors