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Math Help - Vectors - cross and dot products

  1. #1
    Senior Member Educated's Avatar
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    Vectors - cross and dot products

    The unknown vector v satisfies  b \cdot v = \lambda and b \times v = c where lambda, b and c are fixed and known.
    Find v in terms of lambda, b and c

    At first I was thinking of expanding everything out, and so I'd get:

    b_1 v_1 + b_2 v_2 + b_3 v_3 = \lambda
    b_2 v_3 - v_2 b_3, b_3 v_1 - b_1 v_3, b_1 v_2 - b_2 v_1) = (c_1, c_2, c_3)

    But then that didn't seem to help. Any ideas?
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  2. #2
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    Re: Vectors - cross and dot products

    Hey Educated.

    Hint: You have three unknowns (v1,v2,v3) and four equations that are linear. Try setting up a linear system and reducing to get v1,v2,v3 in terms of your other known parameters. Also if there is any chance for in-consistency, then deal with that as well (there shouldn't but you need to check since you have four equations in three unknowns).
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  3. #3
    Senior Member Educated's Avatar
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    Re: Vectors - cross and dot products

    Sorry, I'm not quite sure how to do that. I'll give it a go though:

    b_1 v_1 + b_2 v_2 + b_3 v_3 = \lambda
    b_2 v_3 - b_3 v_2 = c_1
    b_3 v_1 - b_1 v_3 = c_2
    b_1 v_2 - b_2 v_1 = c_3

    \begin{pmatrix} b_1 & b_2 & b_3 & \lambda \\ 0 & -b_3 & b_2 & c_1 \\ b_3 & 0 & -b_1 & c_2\\ -b_2 & b_1 & 0 & c_3 \end{pmatrix}

    I don't know how to reduce it down though. Is there another way of doing this?
    Last edited by Educated; May 29th 2013 at 04:54 AM.
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  4. #4
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    Re: Vectors - cross and dot products

    Personally, I don't like to use matrices to solve systems of equations.
    you have
    b_1v_1+ b_2v_2+ b_3v_3= \lambda
    b_2v_3- b_3v_2= c_1
    b_3v_1- b_1v_3= c_2
    b_1v_2- b_2v_1= c_3

    Multiply the first equation by b_2 to get b_1b_2v_1+ b_2^2v_2+ b_2b_3v_3= b_2\lambda and the second equation by b_3 to get b_2b_3v_3- b_3^2v_2= b_3c_1. Those equations both have " b_2b_3v_3 so subtracting the second fro the first eliminates v_3 giving b_1b_2v_1+(b_2^2+ b_3^2)v_2= b_2\lambda- b_3c_1. Now the fourth equation b_1v_2- b_2v_1= c_3 has only v_1 and v_2 so we can eliminate either of those.

    For example, if we multiply b_1v_2- b_2v_1= c_3 by b_1 we get b_1^2v_2- b_1b_2v_1= b_1c_3 and adding that to b_1b_2v_1+ (b_2^2+ b_3^2)v_2= b_2\lambda- b_2c_1 gives (b_1^2+ b_2^2+ b_3^2)v_2= b_2\lambda+ b_1c_3- b_2c_1[/tex] and can soplve for v_2.
    Thanks from topsquark and Educated
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  5. #5
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    Re: Vectors - cross and dot products

    b.v=λ
    bxv=c

    bx(bxv)=b(b.v)-v(b.b)=bxc
    v=(λb+bxc)(1/b2)
    Thanks from Educated and johng
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  6. #6
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    Re: Vectors - cross and dot products

    If
    v = 2i + 4j
    and
    w = i + 5j
    the
    v . w = (2)(1) + (4)(5) = 22
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  7. #7
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    Re: Vectors - cross and dot products

    Quote Originally Posted by marybalogh View Post
    If
    v = 2i + 4j
    and
    w = i + 5j
    the
    v . w = (2)(1) + (4)(5) = 22
    and 2 + 2 = 4
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