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Calculus II - 31
 

Calculus II - 31

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Stewart Calculus 12.1

Stewart Calculus 12.1

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Calculus II - 31 Calculus II - 31 Presentation Transcript

  • 12.1 3D coordinate systems z (a,b,c) c o b x a y
  • What is the meaning of the followingequation in R ? = = = + + =
  • Distance formula in 3D space:The distance between ( , , ) and ( , , ) is | |= ( ) +( ) +( )An equation of a sphere with center ( , , ) andradius is ( ) +( ) +( ) =
  • 12.2 VectorsA vector is a quantity that has bothmagnitude and direction. B = AThe zero vector has no specific direction.
  • D B v u CA
  • D B v u = C =A
  • D B v u = C =A u
  • D B v u = C =A 2u u
  • D B v u = C =A 2u u u/2
  • D B v u = C =A 2u u -u u/2
  • Vector Addition v u
  • Vector Addition and Subtraction v u u v
  • Vector Addition and Subtraction v u u v The Triangle Law u+v
  • Vector Addition and Subtraction v u u v The Triangle Law u+v u v
  • Vector Addition and Subtraction v u u v The Triangle Law u+v u v
  • Vector Addition and Subtraction v u u v The Triangle Law u+v u u+v The Parallelogram Law v
  • Vector Subtraction v u
  • Vector Subtraction v = +( ) u
  • Vector Subtraction v = +( ) u -v u
  • Vector Subtraction v = +( ) u -v u-v u
  • Properties of vectors: , , are vectors, , are scalars. + = + +( + )=( + )+ + = +( )= ( + )= + ( + ) = + ( ) = ( ) =