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Triple Products

  1. Oct 9, 2014 #1
    1. The problem statement, all variables and given/known data
    [itex] \vec{A}= \hat{i} + 2 \hat{j} \\
    \vec{B}= \hat{j} + \hat{k} \\
    \vec{C} = 2 \hat{k} + \hat{i} [/itex]

    Verify that [itex] \vec{A} \cdot (\vec{B} \times \vec{C}) = \vec{B} \cdot (\vec{C} \times \vec{A}) [/itex]

    2. Relevant equations
    n/a

    3. The attempt at a solution
    I am getting something similar but not equal, I beleive I must have made a simple stupid mistake somewhere, would appreciate a second pair of eyes, thanks :)

    First what I did was rewrite the three vectors like so...
    [itex]
    \vec{A}= (1 \,\, , 2 \,\, , 0) \\
    \vec{B}= (0 \,\, , 1 \,\, , 1) \\
    \vec{C} = (1 \,\, , 0 \,\, , 2) [/itex]

    Then just did [itex] \vec{B} \times \vec{C} [/itex] first...
    [itex]\vec{B} \times \vec{C} = (2-0) \hat{i} - (0-1) \hat{j} + (0-1) \hat{k} = 2 \hat{i} + 1 \hat{j} - 1 \hat{k} = (2 \,\, , 1 \,\, , -1) [/itex]

    And then did the dot product of that new vector with vector A
    [itex]
    \vec{A} \cdot (\vec{B} \times \vec{C}) = (1 \,\, , 2 \,\, , 0) \dot (2 \,\, , 1 \,\, , -1) = (2 \,\, , 2 \,\, , 0)
    [/itex]
    So I got (2, 2, 0) for the left hand side of the equation, now moving onto the right hand side I did the same process.

    First I did just C x A...
    [itex]
    \vec{C} \times \vec{A} = (0-4) \hat{i} - (0-2) \hat{j} + (2-0) \hat{k} = -4 \hat{i} + 2 \hat{j} + 2 \hat{k} = (-4 \,\, , 2 \,\, , 2)
    [/itex]
    And then the dot product of that vector with B
    [itex]
    \vec{B} \cdot (\vec{C} \times \vec{A}) = (0 \,\, , 1 \,\, , 1) \cdot (-4 \,\, , 2 \,\, , 2) = (0 \,\, , 2 \,\, , 2)
    [/itex]

    And as you can see, (0, 2, 2) does not equal (2, 2, 0). Although now thinking about it, the magnitude of the vectors are equal but that is not what the question asked to verify; I was expecting two vectors exactly the same, i.e. (1, 4, 5) and (1, 4, 5), is that what I should get as the answer? Or have I already done it correctly?
     
    Last edited by a moderator: Oct 9, 2014
  2. jcsd
  3. Oct 9, 2014 #2

    DrClaude

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    Staff: Mentor

    You need to review what the dot product is.
     
  4. Oct 9, 2014 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    That's not what the problem was - one of the braces was going the wrong way. It's now fixed in post #1.
     
    Last edited by a moderator: Oct 9, 2014
  5. Oct 9, 2014 #4
    Ah right, of course! The final answer cant be a vector, so they are both just equal to 4?

    I.e.
    [itex]
    \vec{A} \cdot (\vec{B} \times \vec{C}) = (1 \,\, , 2 \,\, , 0) \dot (2 \,\, , 1 \,\, , -1) = 2+2+0 = 4
    [/itex]
     
  6. Oct 10, 2014 #5

    haruspex

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    2016 Award

    Yes.
     
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