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12.2 VectorsA vector is a quantity that has bothmagnitude and direction.                            B                     ...
3D coordinate system:                    z                        (a,b,c)                        c                o       ...
3D coordinate system:                    z                              (a,b,c)                    (a,b,c)                ...
Given points         ( ,   ,   )   and    ( ,   ,   ),the vector       =             ,           ,                      z ...
Given points         ( ,   ,   )   and     ( ,     ,   ),the vector       =             ,            ,                    ...
The addition, subtraction and scalarmultiplication:  ,   ,   +        ,   ,        =            +       ,       +    ,   +...
The addition, subtraction and scalarmultiplication:  ,   ,   +        ,   ,        =            +         ,       +    ,  ...
zStandard basis vectors:                                            (a,b,c) =    , ,                           k =        ...
zStandard basis vectors:                                                 (a,b,c) =    , ,                           k =   ...
The dot product (or inner product) of  =    ,   ,     and       =       ,       ,       is given by           · =         ...
=   ,   ,
=   ,       ,==       +
=             ,           ,      =     =                     +Determinant of order 2:                       =
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Calculus II - 32

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Stewart Calculus 12.2

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Calculus II - 32

  1. 1. 12.2 VectorsA vector is a quantity that has bothmagnitude and direction. B = AThe zero vector has no specific direction.
  2. 2. 3D coordinate system: z (a,b,c) c o b x a y
  3. 3. 3D coordinate system: z (a,b,c) (a,b,c) c o b x a y
  4. 4. Given points ( , , ) and ( , , ),the vector = , , z (x2,y2,z2) (x2-x1,y2-y1,z2-z1) (x1,y1,z1) o x y
  5. 5. Given points ( , , ) and ( , , ),the vector = , , z (x2,y2,z2) (x2-x1,y2-y1,z2-z1) (x1,y1,z1) o x y ! o r 2D Sim ilar f
  6. 6. The addition, subtraction and scalarmultiplication: , , + , , = + , + , + , , , , = , , , , = , ,The length of the vector = , , is | |= + +Unitizing a vector: = | |
  7. 7. The addition, subtraction and scalarmultiplication: , , + , , = + , + , + , , , , = , , , , = , ,The length of the vector = , , is | |= + + ! o r 2DUnitizing a vector: Sim ilar f = | |
  8. 8. zStandard basis vectors: (a,b,c) = , , k = , , i o j x = , , y , , = + +
  9. 9. zStandard basis vectors: (a,b,c) = , , k = , , i o j x = , , y , , = + + ! o r 2D Sim ilar f
  10. 10. The dot product (or inner product) of = , , and = , , is given by · = + +It can also be defined for 2D vectors.The cross product of = , , and = , , is given by = , ,It can only be defined for 3D vectors.
  11. 11. = , ,
  12. 12. = , ,== +
  13. 13. = , , = = +Determinant of order 2: =

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