Geometric Representation of Vectors

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Section 12-1

• Geometric Representation of Vectors

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Vectors

• Vectors are quantities that are described by a
  direction and a magnitude (size). A force would
  be an example of a vector quantity because to
  describe a force, you must specify the direction
  in which it acts and its strength. Velocity is
  another example of a vector.

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Vectors

• The velocities of two airplanes each heading
  northeast at 700 knots are represented by the
  arrows u and v in the diagram on p. 419. We
  write u v to indicate that both planes have the
  same velocity even though the two arrows are
  different.

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Vectors

• In general, any two arrows with the same length
  and the same direction represent the same vector.
  The diagram on p. 419 shows a third airplane
  with speed 700 knots, but because it is heading
  in a different direction, its velocity vector w
  does not equal either u or v.

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Magnitude

• The magnitude of a vector v (also called the
  absolute value of v) is denoted v.

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Addition of Vectors

• If a vector v is pictured by an arrow from point
  A to point B, then it is customary to write v
  . Since the result of moving an object first
  from A to B and then from B to C is the same as
  moving the object directly from A to C, it is
  natural to write . We say that
  is the vector sum of and .

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Addition of Vectors

• The addition of two vectors is a commutative
  operation. In other words, the order in which
  the vectors are added does not make any
  difference. You can see this in the diagrams on
  p. 420 where the red arrows denote a b and b
  a having the same length and direction.

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Addition of Vectors

• If the two diagrams on p. 420 are moved together,
  a parallelogram is formed. This suggests that
  another way to add a and b is to draw a
  parallelogram OACB with sides a and
  b. The diagonal of the parallelogram
  is the sum.
• This method is frequently used in physics
  problems involving forces that are combined.

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Vector Subtraction

• The negative of a vector v, denoted v, has the
  same length as v but the opposite direction. The
  sum of v and v is the zero vector 0.
• It is best thought of as a point.
• v (-v) 0

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Vector Subtraction

• Vectors can be subtracted as well as added.
• v w means v (-w).

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Multiples of a Vector

• The vector sum v v is abbreviated as 2v.
  Likewise, v v v 3v. The diagram on p. 421
  shows that the arrows representing 2v and 3v have
  the same direction as the arrow representing v,
  but that they are two and three times as long.

12
Multiples of a Vector

• In general if k is a positive real number, then
  kv is the vector with the same direction as v but
  with an absolute value k times as large. If k lt
  0, then kv has the same direction as v and has
  an absolute value k times as large. If k ? 0,
  then is defined to be equal to the vector
  .

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Scalars

• When working with vectors, it is customary to
  refer to real numbers as scalars.

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Scalar Multiplication

• When this is done, the operation of multiplying a
  vector v by a scalar k is called scalar
  multiplication. This operation has the following
  properties. If v and w are vectors and k and m
  are scalars, then
• k(v w) kv kw
• (k m)v kv mv
• k(mv) (km)v Associative law

Distributive laws
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True or False?
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Complete the statement.
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Homework p. 423-424 1-9 odd
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Homework p. 423-424 1-9 odd
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Homework p. 423-424 1-9 odd
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Homework p. 423-424 1-9 odd
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Homework p. 423-424 1-9 odd

• 9. A ship travels 200 km west from port and then
  240 km due south before it is disabled.
  Illustrate this in a vector diagram. Use
  trigonometry to find the course that a rescue
  ship must take from port in order to reach the
  disabled ship.