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12.4 The Cross Product The cross product of       =   ,   ,       and   =    ,   ,   is given by       =                , ...
Properties (I):         =        ( + )=      +(   )        = (   )=         ( )         =                        · = ·    ...
Properties (II):      =|    | = | || |     =                   ·     =| |                       · = | || |                ...
Properties (III):(     )     ,(        )|     | = | || |          equals to the area ofthe parallelogram determined by    ...
Properties (IV):  ·(      )=(          )·  ·(      )   is called the scalar triple product              of   , , .  ·(    ...
Properties (V):    (      )=( · )    ( · )    (      )=(    )
Properties (V):         (        )=( · )         ( · )         (        )=(    )Properties (I-IV):           =         ( +...
12.5 Equations of Lines      and Planes Vector equation of a line:                        =       + If   =    , ,      ,  ...
Vector equation of a line:                       =       +If   =    , ,      ,       =       ,       ,       ,   =   , ,  ...
Ex: Find an equation of the line pass throughtwo given points ( , ,   ) and ( , , )Ex: Show that the lines with parametric...
Vector equation of a plane:                         ·(       )=If    =        , ,   ,        =   ,    ,       ,   =    , ,...
Ex: Find an equation of the plane throughthe point ( , ,   ) with normal vector ,        ,Ex: Find an equation of the plan...
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Calculus II - 34

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Stewart Calculus 12.4&5

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Calculus II - 34

  1. 1. 12.4 The Cross Product The cross product of = , , and = , , is given by = , , = = + It can only be defined for 3D vectors.
  2. 2. Properties (I): = ( + )= +( ) = ( )= ( ) = · = · · )= · + ·( + ·( ) = ( · )= ( )· · =
  3. 3. Properties (II): =| | = | || | = · =| | · = | || | · =
  4. 4. Properties (III):( ) ,( )| | = | || | equals to the area ofthe parallelogram determined by and . The Right Hand Rule: If the fingers of your right hand curl in the direction of a rotation from to , then your thumb points in the direction of .
  5. 5. Properties (IV): ·( )=( )· ·( ) is called the scalar triple product of , , . ·( )=The volume of the parallelepipeddetermined by the vectors , , equals | ·( )|.
  6. 6. Properties (V): ( )=( · ) ( · ) ( )=( )
  7. 7. Properties (V): ( )=( · ) ( · ) ( )=( )Properties (I-IV): = ( + )= +( ) = ( )= ( ) = =| | = | || | =( ) ,( ) ·( )=( )·
  8. 8. 12.5 Equations of Lines and Planes Vector equation of a line: = + If = , , , = , , , = , , , then , , = + , + , + Parametric equation: = + , = + , = +
  9. 9. Vector equation of a line: = +If = , , , = , , , = , , ,then , , = + , + , +Parametric equation: = + , = + , = +symmetric equation: = =
  10. 10. Ex: Find an equation of the line pass throughtwo given points ( , , ) and ( , , )Ex: Show that the lines with parametricequations = + , = + , = = , = + , = +do not intersect and are not parallel.
  11. 11. Vector equation of a plane: ·( )=If = , , , = , , , = , , ,then: , , · , , =Scalar equation: ( )+ ( )+ ( )=Linear equation: + + + =
  12. 12. Ex: Find an equation of the plane throughthe point ( , , ) with normal vector , ,Ex: Find an equation of the plane that passesthrough ( , , ), ( , , ) and ( , , ).Ex: Find the point at which the line withparametric equations = + , = + , =intersects the plane + = .Ex: Find a equation for the line ofintersection of two planes = , + = .

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