This Week

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This Week
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Short Course in Mathematics and Analytic Geometry

• Week 2
• Trigonometry and Vectors

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Radians

• A circle of radius equal to one is called a unit
  circle.
• On a unit circle, an angle measured in radians is
  defined as the arc length this angle subtends.
• Anticlockwise is
• positive radians
• Clockwise is
• negative radians

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Arc Length and Area of a Sector

• In a regular circle, the arc-length is given as
• The area of a sector is given as

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Trigonometric Functions

• Last week we looked at the three main
  trigonometric functions

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Trigonometric Inverse Functions

• In addition to these three functions are their
  inverse functions

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Trigonometry of Lines

• Consider the catheti used to determine the
  Euclidean distance between points A and B
• This implies the following relationship for
  gradients

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Trigonometry of Circles

• In a circle, the trigonometric functions have the
  following relationships

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Trigonometric Identities

• Consider the sine and cosine relationships about
  a unit circle and the equation for a unit circle
  with its centre at the origin
• Substituting items on the left into the equation
  on the right gives the trigonometric identity
• Additional trigonometric identities
• http//www.members.optusnet.com.au/xyzrune/THQLect
  ures/images/TrigonometricIdentities.pdf

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The Law of Sines and Cosines

• Consider a general triangle with angles A, B, C
  and corresponding sides a, b, c
• The law of
  sines
• The law of
  cosines
  

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Vectors (Part 1)

• Scalars A scalar is a variable that only has
  magnitude. It does not relate to any definite
  direction in space.
• Vectors A vector is a variable that has both
  magnitude and direction

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Vector Space

• A vector space is not an Euclidean space.
• A vector space can map onto an R2 space if two
  special vectors i and j, called standard basis
  vectors, are allowed to represent a unit of
  magnitude along each axis, x and y

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A Vector on R2

• Given standard basis vectors i and j on an R2
  space, any vector v can be defined in terms these
  standard basis vectors. For example

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A Vector on 3D

• As with Euclidean space, a vector space can be
  extended to higher dimensions

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Properties of Vectors

• Given two vectors
• Vector v is equal to w if and only if
• Given a is a scalar, the scalar product of a
  vector v is

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Properties of Vectors

• The sum of vectors v and w is

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Properties of Vectors

• The difference between vectors v and w is

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Magnitude of a Vector

• The magnitude of a vector v is given by the
  notation v .
• On a Euclidean space, the magnitude is given by
  the Euclidean distance formula

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Unit Vectors

• Quite often the direction of a vector is more
  important than the magnitude. Any vector can be
  generalized to a vector of unit length, called a
  unit vector
• That is

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Standard Basis Vectors

• Properties of the Standard Basis vectors

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Dot Product

• Note The dot product of two vectors is always a
  scalar.

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Angle between Vectors

• The dot product may be used to calculate the
  angle between two non-zero vectors

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Orthogonal Vectors

• If the angle between two non-zero vectors is ?/2
  then the vectors are said to be orthogonal

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Next Week

• Vectors (Part 2)