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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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