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STUPID Vector qusetion - dont understand dot product rule

  1. Jan 18, 2009 #1
    1. The problem statement, all variables and given/known data

    The points A and B have position vectors a = (2,2,1) and b (1,1,-4) respectively relative to an origin O. (im using column notation for shorthand)

    Prove that OA is perpendicular to AB

    2. Relevant equations

    3. The attempt at a solution

    To be perpendicular the angle between the lines is 90

    using the dot product rule:

    vectors a and b should multiply to give cos 90 (which is 0)

    AB = (3,3,-3)-(2,2,1) = (1,1,-4)
    (2,2,1).(1,1,-4) = (2,2,-4)

    NOW WHAT I DONT UNDERSTAND IS why you add the components i,j,k to get 0. I can see 2 + 2 -4 =0, but why do you do this? Why can you do this

    Thanks :)
  2. jcsd
  3. Jan 18, 2009 #2
    Simple answer: A dot B is a number (scalar) - not a vector. It's definition is: axbx + ayby + azbz.
  4. Jan 18, 2009 #3


    Staff: Mentor

    You're using row notation. Columns are vertical.
    Are you sure you have written the problem correctly? Vectors OA and OB are perpendicular, but OA and AB aren't.
    Where did you get (3, 3, 3)? Vector AB = OB - OA, which is (1, 1, -4) - (2, 2, 1) = (-1, -1, -5).
  5. Jan 18, 2009 #4
    It's just ordered-set notation; neither row nor column vectors have commas!
  6. Jan 19, 2009 #5
    You did show that OA is perpendicular to OB. That's good.

    I think your question is "Why does the dot-product rule work? Why do you multiply similar components, and then add up the sum?"

    Here is one answer that might help you make sense of it:
    Look at the dot-product of A.A: it is simply the Pythagorean Theorem.
    If A = (ax, ay, az)
    Then A.A = ax2 + ay2 + az2.
    So ... the dot-product is a method of determining the length of a vector: Lth (A) = [tex]\sqrt{A.A}[/tex]

    That's just an example to make you feel comfortable with the rule - because it's useful and makes sense in that situation.

    Good luck.

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