Chapter 2. Vector Analysis 22, 23, Vector Algebra pp. 1119

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Chapter 2. Vector Analysis2-2, 2-3, Vector
Algebra (pp. 11-19)

• Scalar has only magnitude (time, mass,
  distance) A,B
• Vector has both magnitude and direction
  (velocity, force) A, B or A, B
• Field function that specifies a particular
  quantity everywhere in a region.
• Scalar field electric potential distribution
• Vector field velocity of raindrops
• Vector magnitude A, A
• Unit vector aA has a unit magnitude and
  direction along A

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• In Cartesian coordinates
• Ax Ay Az are called the components of A in the
  x, y, z directions.
• ax ay az are unit vectors in the x, y, z
  directions.
• The magnitude of vector A is given by
• and the unit vector along A is given by

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• Vector additions
• Basic laws
• commutative
• associative
• distributive

4

• Example
• 1)
• 2)
• 3) A unit vector along
• Position and Distance vectors
• P (x, y, z) is a point in Cartesian coordinates
• Position vector

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• Distance vector
• Example P (0, 2, 4) Q (-3, 1, 5)
• 1) Position vector P
• 2) Distance vector from P to Q
• 3) Distance between P and Q
• 4) A vector parallel to PQ with magnitude of 10
• Example Sketch the vector field

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• 2-3 Vector Multiplication
• There are two types of vector multiplication
• 1) Scalar (dot) product
• 2) Vector (cross) product
• plus
• 3) Scalar triple product
• 4) Vector triple product
• Dot Product
• If

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• then
• The angle between A and B can be calculated as
• The dot product obeys
• Commutative law
• Distributive law

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• Cross Product
• 1) right-handed screw rule
• 2) right-hand rule
• if
• then

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• Determining the component of a vector
• The scalar component AB of A along vector B
• The vector component AB
• A - AB is perpendicular to B
• Example
• find the angle between A and B

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• Cross Product Properties
• 1) anticommutative
• 2) it is not associative
• 3) distributive
• Scalar Triple Product

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• if
• then
• the result is a scalar, it is the volume of a
  parallelepiped.
• Vector Triple Product

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• Example
• find
• 1)
• 2)
• 3)
• 4)
• 5)
• 6) A unit vector perpendicular to both Q and R
• 7) The component of P along Q