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Dot Product Clarification (Kleppner & Kolenkow p.9)

  1. Jun 4, 2013 #1
    Problem:

    In Kleppner's book, Introduction to Mechanics, he states

    "By writing [itex]\vec{A}[/itex] and [itex]\vec{B}[/itex] as the sums of vectors along each of the coordinate axes, you can verify that [itex]\vec{A} \cdot \vec{B} = A_{x}B_{x} + A_{y}B_{y} + A_{z}B_{z}[/itex]."

    He suggests summing vectors, but since the sum of two vectors vectors [itex]\vec{A}[/itex] and [itex]\vec{B}[/itex] is a new vector [itex]\vec{C}[/itex], I don't understand how the result could be a scalar. Am I missing something?

    When I was introduced to the dot product in Stewart's Calculus, he presents it as definition.
     
  2. jcsd
  3. Jun 4, 2013 #2
    I don't get it either. What is the context in which Kleppner makes this statement?
     
  4. Jun 4, 2013 #3

    Doc Al

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    Staff: Mentor

    He's not suggesting summing vectors but representing each vector as the sum of its components times unit vectors.

    Like:
    A = Ax i + Ay j + Az k

    Then you can take the scalar product (of A and B) and see the result more easily.
     
  5. Jun 4, 2013 #4

    CAF123

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    Gold Member

    I have the book as well. Does he not simply mean write: $$\mathbf{A} = A_x \hat{x} + A_y \hat{y} + A_z \hat{z}\,\,\,\,\text{and}\,\,\,\, \mathbf{B} = B_x \hat{x} + B_y \hat{y} + B_z \hat{z}$$ and then take dot product.

    I assume by writing vectors A and B like this is what he means by 'sum of vectors along each of the coordinate axes'.

    Edit: Doc Al said same thing.
     
  6. Jun 4, 2013 #5
    The above are correct
     
    Last edited: Jun 4, 2013
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