MAC 1114

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MAC 1114

• Module 9
• Introduction to Vectors

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Learning Objectives

• Upon completing this module, you should be able
  to
• Learn and apply basic concepts about vectors.
• Perform operations on vectors.
• Represent a vector quantity algebraically and
  find unit vectors.
• Compute dot products and the angle between two
  vectors.
• Use vectors to solve applications.

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Introduction to Vectors
There are two major topics in this module
- Introduction to Vectors, Operations, and the
Dot Products - Application of Vectors
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Quick Review on Parallel Lines and Transversal

• Parallel lines are lines that lie in the same
  plane and do not intersect.
• When a line q intersects two parallel lines, q,
  is called a transversal.

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Important Angle Relationships
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Basic Terminology

• A vector in the plane is a directed line segment.
  
• Consider vector AB
• A is called the initial point
• B is called the terminal point
• Magnitude length of a vector, expressed as
• The sum of two vectors is also a vector.
• The vector sum A B is called the resultant.

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Basic Terminology Continued

• A vector with its initial point at the origin is
  called a position vector.
• A position vector u with its endpoint at the
  point (a, b) is written
• The numbers a and b are the horizontal component
  and vertical component of vector u.
• The positive angle between the x-axis and a
  position vector is the direction angle for the
  vector.

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What are Magnitude and Direction Angle of Vector
?

• The magnitude (length) of vector u is
  given by
• The direction angle ? satisfies
  where a ? 0.

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Example of Finding Magnitude and Direction Angle

• Find the magnitude and direction angle for
• Magnitude
• Direction Angle

• Vector u has a positive horizontal component.
• Vector u has a negative vertical component,
  placing the vector in quadrant IV.

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What are the Horizontal and Vertical Components?

• The horizontal and vertical components,
  respectively, of a vector u having magnitude u
  and direction angle ? are given by
• That is,

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Example of Finding the Horizontal and Vertical
Components

• Vector w has magnitude 35.0 and direction angle
  51.2. Find the horizontal and vertical
  components.
• Therefore, w
• The horizontal component is 21.9, and the
  vertical component is 27.3.

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Example

• Write each vector in the Figure on the right in
  the form

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Solutions

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What are Vector Operations?

• For any real numbers a, b, c, d, and k,

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Example Vector Operations
find
and
Let

• a) 4v
• b) 2u v

• c) 2u - 3v

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How to Compute Dot Product?

• A unit vector is a vector that has magnitude 1.
• Dot Product
• The dot product of two vectors
• is
  denoted u v, read u dot v, and given by

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Example of Finding Dot Products

• Find each dot product.

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What are the Properties of the Dot Product?

• For all vectors u, v, and w and real numbers k,
• a)
• b)
• c)
• d)
• e)
• f)

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What is the Geometric Interpretation of Dot
Product?

• If ? is the angle between the two nonzero vectors
  u and v, where 0 ? 180, then

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Example of Finding the Angle Between the Two
Vectors

• Find the angle ? between two vectors
• By the geometric

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Example

• Forces of 10 newtons and 50 newtons act on an
  object at right angles to each other. Find the
  magnitude of the resultant and the angle of the
  resultant makes with the larger force.
• The resultant vector, v, has magnitude 51 and
  make an angle of 11.3 with the larger force.

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Example

• A vector w has a magnitude of 45 and rests on an
  incline of 20. Resolve the vector into its
  horizontal and vertical components.
• The horizontal component is 42.3 and the vertical
  component is 15.4.

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Example

• A ship leaves port on a bearing of 28.0 and
  travels 8.20 mi. The ship then turns due east and
  travels 4.30 mi. How far is the ship from port?
  What is its bearing from port?

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Example Continued

• Vectors PA and AE represent the ships path.
• Magnitude and bearing

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Example Continued

• The ship is about 10.9 mi from port.
• To find the bearing of the ship from port, find
  angle APE.
• Add 20.4 to 28.0 to find that the bearing is
  48.4.

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What is the Equilibrant?

• We have learned how to find the resultant of two
  vectors.
• A vector that will counterbalance the resultant
  is called the equilibrant. For instance, the
  equilibrant of vector u is the vector -u.

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What have we learned?

• We have learned to
• Learn and apply basic concepts about vectors.
• Perform operations on vectors.
• Represent a vector quantity algebraically and
  find unit vectors.
• Compute dot products and the angle between two
  vectors.
• Use vectors to solve applications.

http//faculty.valenciacc.edu/ashaw/ Click link
to download other modules.
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Credit

• Some of these slides have been adapted/modified
  in part/whole from the slides of the following
  textbook
• Margaret L. Lial, John Hornsby, David I.
  Schneider, Trigonometry, 8th Edition

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