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Math Help - Vector problem

  1. #1
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    Vector problem

    Find (\vec{a}\times \vec{b})\cdot \vec{c} if \vec{a}=3\vec{m}+5\vec{n}, \vec{b}=\vec{m}-2\vec{n}, \vec{c}=2\vec{m}+7\vec{n}, |\vec{m}|=\frac{1}{2}, |\vec{n}|=3 and angle between \vec{m} and \vec{n} is \frac{3\pi}{4}.

    I've tried to substitute [(3\vec{m}+5\vec{n})\times({\vec{m}-2\vec{n})]\cdot(2\vec{m}+7\vec{n}) and by doing a cross product leading me to the expression (11\vec{n}\times\vec{m})\cdot(2\vec{m}+7\vec{n}).

    My book gives this solution: (\vec{a}\times \vec{b})\cdot \vec{c}=0.

    Thank you
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: Vector problem

    Quote Originally Posted by patzer View Post
    Find (\vec{a}\times \vec{b})\cdot \vec{c} if \vec{a}=3\vec{m}+5\vec{n}, \vec{b}=\vec{m}-2\vec{n}, \vec{c}=2\vec{m}+7\vec{n}, |\vec{m}|=\frac{1}{2}, |\vec{n}|=3 and angle between \vec{m} and \vec{n} is \frac{3\pi}{4}.

    I've tried to substitute [(3\vec{m}+5\vec{n})\times({\vec{m}-2\vec{n})]\cdot(2\vec{m}+7\vec{n}) and by doing a cross product leading me to the expression (11\vec{n}\times\vec{m})\cdot(2\vec{m}+7\vec{n}).

    My book gives this solution: (\vec{a}\times \vec{b})\cdot \vec{c}=0.

    Thank you
    \vec{a}-\vec{b}=\vec{c}

    So, \vec{a}, \vec{b}, \vec{c} are linearly dependent and their triple product (\vec{a}\times \vec{b})\cdot \vec{c}=0.
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  3. #3
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    Re: Vector problem

    Quote Originally Posted by alexmahone View Post
    \vec{a}-\vec{b}=\vec{c}

    So, \vec{a}, \vec{b}, \vec{c} are linearly dependent and their triple product (\vec{a}\times \vec{b})\cdot \vec{c}=0.
    That's ok, but is there any other way to solve it. I mean, imagine if \vec{c} is another vector p.s \vec{c}=2\vec{m}+6\vec{n}.
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  4. #4
    MHF Contributor alexmahone's Avatar
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    Re: Vector problem

    Quote Originally Posted by patzer View Post
    That's ok, but is there any other way to solve it. I mean, imagine if \vec{c} is another vector p.s \vec{c}=2\vec{m}+6\vec{n}.
    In that case I would do it the way you tried in post #1.

    PS: On second thoughts, the vectors would still be linearly dependent and so their triple product would be zero.
    Last edited by alexmahone; August 24th 2011 at 06:29 AM.
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  5. #5
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    Re: Vector problem

    Sir, it was an attempt and didn't know what to do there. Can you write the solution for me so I can see where I did it wrong.
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  6. #6
    MHF Contributor alexmahone's Avatar
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    Re: Vector problem

    Quote Originally Posted by patzer View Post
    Sir, it was an attempt and didn't know what to do there. Can you write the solution for me so I can see where I did it wrong.
    I don't know why you're trying to complicate matters when I've given you a simple solution in post #2.
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  7. #7
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    Re: Vector problem

    I don't know why you're trying to complicate matters even more, while I've asked a simple question in the first post. I said, given these facts, help me to solve algebraicaly this problem.
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  8. #8
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    Re: Vector problem

    Again, this is my approach to this problem:

    (\vec{a}\times\vec{b})\cdot\vec{c}=[(3\vec{m}+5\vec{n})\times(\vec{m}-2\vec{n})]\cdot(2\vec{m}+7\vec{n}) =
    =[3\vec{m}\times\vec{m}-6\vec{m}\times\vec{n}+5\vec{n}\times\vec{m}-10\vec{n}\times\vec{n}]\cdot(2\vec{m}+7\vec{n})

    [(\vec{m}\times\vec{m}=0, \vec{n}\times\vec{n}=0, (\vec{m}\times\vec{n})=-(\vec{n}\times\vec{m})]

    so... (-11\vec{m}\times\vec{n})\cdot(2\vec{m}+7\vec{n}) I stopped here because I didn't know what to do.
    So, here I need your help on this. I appreciate the alexmahone's help, but I'm looking to solve the problem using the facts that were mentioned in problem.
    Last edited by patzer; August 24th 2011 at 09:43 AM.
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  9. #9
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    Re: Vector problem

    Quote Originally Posted by patzer View Post
    Again, this is my approach to this problem:

    (\vec{a}\times\vec{b})\cdot\vec{c}=[(3\vec{m}+5\vec{n})\times(\vec{m}-2\vec{n})]\cdot(2\vec{m}+7\vec{n}) =
    =[3\vec{m}\times\vec{m}-6\vec{m}\times\vec{n}+5\vec{n}\times\vec{m}-10\vec{n}\times\vec{n}]\cdot(2\vec{m}+7\vec{n})

    [(\vec{m}\times\vec{m}=0, \vec{n}\times\vec{n}=0, (\vec{m}\times\vec{n})=-(\vec{n}\times\vec{m})]

    so... (-11\vec{m}\times\vec{n})\cdot(2\vec{m}+7\vec{n}) I stopped here because I didn't know what to do.
    So, here I need your help on this. I appreciate the alexmahone's help, but I'm looking to solve the problem using the facts that were mentioned in problem.
    Try distributing the cross product over the sum.
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  10. #10
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    Re: Vector problem

    Hello Ackbeet,

    I did these operations: (-11\vec{m}\times\vec{n})\cdot(2\vec{m}+7\vec{n})=(-11\vec{m}\times\vec{n})\cdot2\vec{m}+(-11\vec{m}\times\vec{n})\cdot7\vec{n} =
    = -22(\vec{m}\times\vec{n})\cdot\vec{m}-77(\vec{m}\times\vec{n})\cdot\vec{n}

    So, according to definition, the vector \vec{m}\times\vec{n} is normal with the plane spanned by vectors \vec{m} and \vec{n}, as a result we get 0 at the and. Is this correct?
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  11. #11
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    Re: Vector problem

    Quote Originally Posted by patzer View Post
    Hello Ackbeet,

    I did these operations: (-11\vec{m}\times\vec{n})\cdot(2\vec{m}+7\vec{n})=(-11\vec{m}\times\vec{n})\cdot2\vec{m}+(-11\vec{m}\times\vec{n})\cdot7\vec{n} =
    = -22(\vec{m}\times\vec{n})\cdot\vec{m}-77(\vec{m}\times\vec{n})\cdot\vec{n}

    So, according to definition, the vector \vec{m}\times\vec{n} is normal with the plane spanned by vectors \vec{m} and \vec{n}, as a result we get 0 at the and. Is this correct?
    Sounds good to me, assuming all your work up until that point is correct, which I haven't checked.
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