Sec 13.3The Dot Product

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Sec 13.3 The Dot Product

• Definition
• The dot product is sometimes called the scalar
  product or the inner product of two vectors.

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Properties of the Dot Product

• If a, b, and c are vectors, and k is a scalar,
  then

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Theorem If ? is the angle between the vectors a
and b, then a b ab cos ?

• Corollary If ? is the angle between the
  non-zero vectors
• a and b, then
• Definition Perpendicular vectors are also
  called
• orthogonal vectors.
• Corollary Two vectors a and b are orthogonal
  if and
• only if a b 0

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Direction Angles and Direction Cosines

• The direction angles of a non-zero vector a are
  the angles a, ß, and ? (in the interval 0, p)
  that a makes with the positive x-, y-, and
  z-axes.
• The cosines of these direction angles cos a,
  cos ß, and
• cos ? are called the direction cosines of the
  vector a.
• Theorem

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Vector and Scalar Projections

• The vector projection of b onto a is denoted and
  defined by
• The scalar projection of b onto a (also called
  the component of b along a) is defined to be the
  signed magnitude of the vector projection, which
  is the number
• b cos ? , where ? is the angle between a and b.
  

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Sec 13.4 The Cross Product

• Definition
• Note
• The cross product is defined only for
  three-dimensional vectors.

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Theorem The vector a b is orthogonal to both
a and b.

• Theorem
• If ? is the angle between a and b (so 0 ?
  p), then
• a b a b sin ?
• Corollary Two non-zero vectors a and b are
  parallel if
• and only if a b 0
• Theorem The length of the cross product a b
  is equal
• to the area of the
  parallelogram determined by
• a and b.

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Theorem i j k j k i
k i j j i -k k
j - i i k -j

• Theorem If a, b, and c are vectors and k is a
  scalar, then
• 1. a b - b a
• (ka) b k(a b) a (kb)
• 3. a (b c) a b a c
• 4. (a b) c a c b c
• 5. a (b c) (a b) c
• 6. a (b c) (a c) b - (a b) c
• Note The cross product is neither commutative
  nor associative.

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Triple Products

• Definition
• Theorem The volume of the parallelepiped
  determined
• by the vectors a, b, and c is the magnitude of
• their scalar product V a (b c)