Equality of vectors Multiplication by a scalar Addition Vector addition is commutative Vector additi

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Equality of vectors
Multiplication by a scalarAdditionVector
addition is commutativeVector addition is
associativeAddition and subtraction
Review Vector Equality and Addition
Remember our motto?
REPETITIO MATER STUDIORUM EST
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Is our Algebra Sufficient?

• What else do we need besides vector addition and
  scalar multiplication?
• Incomplete because it fails to indicate the
  difference between scalars and vectors
• The distinction still resides only in their
  geometric interpretation
• The opportunity to give the notion of direction a
  full algebraic expression arises when we consider
  how to multiply vectors.

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Relative Orientation Between Two Vectors

• First distinguish between the length and
  direction

Design product between the vectors to include
both
Product of lengths Information on relative
orientation
There are two simple, single valued choices for
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Scalar Product of Vectors

• Use it to
• Express the fact that two vectors are orthogonal
• Find the magnitude of a vector
• Find the angle between two unit vectors
• Find the angle between two vectors
• Find the projection of a vector onto a given
  direction
• Express theorems of triangle geometry

Scalar product is commutative
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Algebra at Work Cosine Theorem
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Vector Product of Vectors
sense defined by the right-hand-rule

• Use it to
• Express the fact that two vectors are parallel
• Find the area of a parallelogram
• Find the component of a vector orthogonal to a
  given direction
• Express theorems of triangle geometry

Vector product is anti-commutative!
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Algebra at Work Misc.
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Algebra at Work Sine Theorem