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12.3 The Dot ProductThe dot product (or inner product) of  =    ,     ,           and       =       ,       ,       is giv...
Properties (I):    · = ·    ·( + )=       · +   ·(       )· = ( · )=     ·( )    ·    =
Properties (II):  ·   =| |  · = | || |  · =
Properties (II):  ·   =| |  · = | || |  · =  · >              · =   · <
Direction Angles and Direction CosinesThe direction angles of a nonzero vector arethe angles that it makes with the positi...
Projections:The scalar projection of onto (also calledthe component of onto ) is defined to be                            ...
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       a...
Properties (III):(     )     ,(        )|     | = | || |          equals to the area ofthe parallelogram determined by    ...
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Calculus II - 33

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Stewart Calculus 12.3&4

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Calculus II - 33

  1. 1. 12.3 The Dot ProductThe dot product (or inner product) of = , , and = , , is given by · = + +It can also be defined for 2D vectors.Ex: , , · , , = · + ( )+ · =+ · = · + · +( )( )=
  2. 2. Properties (I): · = · ·( + )= · + ·( )· = ( · )= ·( ) · =
  3. 3. Properties (II): · =| | · = | || | · =
  4. 4. Properties (II): · =| | · = | || | · = · > · = · <
  5. 5. Direction Angles and Direction CosinesThe direction angles of a nonzero vector arethe angles that it makes with the positive x-,y-, and z-axes. · · = = = = | || | | | | || | | | · = = | || | | | + + = =| | , ,
  6. 6. Projections:The scalar projection of onto (also calledthe component of onto ) is defined to be · =| | = calar! | | sig ne d sThe projection of onto is defined to be · · = = ve ctor! | | | | | |
  7. 7. 12.4 The Cross Product The cross product of = , , and = , , is given by = , , = = + It can only be defined for 3D vectors.
  8. 8. Properties (I): = ( + )= +( ) = ( )= ( ) = · = · · )= · + ·( + ·( ) = ( · )= ( )· · =
  9. 9. Properties (II): =| | = | || | = · =| | · = | || | · =
  10. 10. Properties (III):( ) ,( )| | = | || | equals to the area ofthe parallelogram determined by and .
  11. 11. 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 .

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