# Chapter 6 - Chapter Chapter 6 Three Dimensional Space...

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hapter 6 Chapter 6 Three Dimensional Space

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Overview he Cartesian Coordinate System The Cartesian Coordinate System ectors Vectors Definitions l b t 2 t Angle between 2 vectors Scalar or Dot Product roperties of Scalar Product Properties of Scalar Product
Overview nit Vectors Unit Vectors Projection Vector Product roperties of Vector Product Properties of Vector Product ines in 3 Space Lines in 3-D Space Vector Equation of a Line arametric Equation of a Line Parametric Equation of a Line

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Overview lanes Planes Equation for Plane istance from Point to Plane Distance from Point to Plane ector Functions of One Variable Vector Functions of One Variable Limits and Continuity erivatives of Vector Functions Derivatives of Vector Functions Definite Integral of a Vector Function
Overview pace Curves Space Curves Smooth Curves angent Vector and Line to a Curve Tangent Vector and Line to a Curve Arc Length of a Space Curve

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he Cartesian Coordinate The Cartesian Coordinate System
The Cartesian Coordinate System ectangular Coordinates Rectangular Coordinates c z (,,) Pabc Right-handed coordinates system b y 0 a x If we rotate the axis counterclockwise toward the axis, xy  then a right-handed screw will move in the positive direction. z

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The Cartesian Coordinate System 0 , 0 , 0 : Planes z y x z Eight Octants 0 y 0 z 0 x x y
The Cartesian Coordinate System istane between two points Distane between two points z 2222 (, , ) Px y z 111 , ) y z 1111 y 0 x 2 1 2 2 1 2 2 1 2 2 1 ) ( ) ( ) ( z z y y x x P P

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The Cartesian Coordinate System 00 Equation of circle of radius and center ( , ) rx y x r ) y ( , xy 2 2 0 y 22 Equation of circle: () ( ) x x yy r 
The Cartesian Coordinate System 00 0 Equation of the sphere of radius and center ( , , ) rx y z z r ) yz 0 000 ( ,, xyz x y 2 2 0 2 0 2 0 ) ( ) ( ) ( : sphere of Equation r z z y y x x

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Vectors
Vectors – Definitions directed line segment PQ A directed line segment PQ Q (terminal point) (initial point) P Direction: Magnitude: direction of the arrow length of the line segment Two vectors are equal if they have the same direction and length. qy g =

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Vectors – Definitions he vector of ( ) : x yz osition 0 x   00 0 The ,, ) : Px position 0 0 OP y z   The of is denoted by : OP length 22 2 0 || || OP x y z  () magnitude z 0 0 (, , ) Px y z . 0 0 is vector The 0 zero 0 x y
Vectors – Definitions 2 x Addition 1 1 1 y x v 2 2 2 z y v 1 z 1 v 2 1 2 1 2 1 y y x x v v 2 v 2 1 z z

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Vectors – Definitions 2 x Addition 1 1 1 y x v 2 2 2 z y v 1 z 2 v 1 v 2 1 2 1 2 1 y y x x v v 2 1 z z
Vectors – Definitions Negative

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Chapter 6 - Chapter Chapter 6 Three Dimensional Space...

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