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Math Help - Find the vector 'b'

  1. #1
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    Find the vector 'b'

    Question :

    Find a vector b such that a.b = 1 , a \times b = j - k and a = i + j + k
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  2. #2
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    Quote Originally Posted by zorro View Post
    Question :

    Find a vector b such that a.b = 1 , a \times b = j - k and a = i + j + k
    Let b = ui + vj + wk.

    Substitute it and a = i + j + k into a.b = 1 and a \times b = j - k.

    Get some equations and use them to solve for u, v and w.

    If you need more help, please show all of your working and state specifically where you're still stuck.
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  3. #3
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    I am so sorry but i am stull stuck there

    Quote Originally Posted by mr fantastic View Post
    Let b = ui + vj + wk.

    Substitute it and a = i + j + k into a.b = 1 and a \times b = j - k.

    Get some equations and use them to solve for u, v and w.

    If you need more help, please show all of your working and state specifically where you're still stuck.

    I am so sorry but i am still stuck there i tried to do what u adviced me to do but it s not leading me anywhere

    This is what i have done till now!!!

    b = ui + vj + wk ;  a = i + j + k

     a.b = 1 and  a \times b = j - k

     a.b = (i + j + k)(ui + vj + wk) = ui^2 + vj^2 + wk^2

    Now what should i do with a.b

    And
     a \times b = \begin{vmatrix} i & j & k \\ 1 & 1 & 1 \\ u & v & w \end{vmatrix} =  i(v - w) - j(w - u) + k(v - u)

    therefore
     i(v - w) - j(w - u) + k(v - u) = j - k

    what should i do with this???
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  4. #4
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    Quote Originally Posted by zorro View Post
    I am so sorry but i am still stuck there i tried to do what u adviced me to do but it s not leading me anywhere

    This is what i have done till now!!!

    b = ui + vj + wk ;  a = i + j + k

     a.b = 1 and  a \times b = j - k

     a.b = (i + j + k)(ui + vj + wk) = ui^2 + vj^2 + wk^2 Mr F says: You're expected to know that i^2 = j^2 = k^2 = 1. Therefore ....

    Now what should i do with a.b

    And
     a \times b = \begin{vmatrix} i & j & k \\ 1 & 1 & 1 \\ u & v & w \end{vmatrix} =  i(v - w) - j(w - u) + k(v - u)

    therefore
     i(v - w) - j(w - u) + k(v - u) = j - k Mr F says: Equate components on each side. eg. Equating i components gives v - w = 0.

    what should i do with this???
    Use the resulting four equations to solve for u, v and w (and hence b).
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  5. #5
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    Is the answer correct ?

    Quote Originally Posted by mr fantastic View Post
    Use the resulting four equations to solve for u, v and w (and hence b).

    Is this correct ?

    a.b = u + v + w

    And since
    Mr F says: Equate components on each side. eg. Equating i components gives v - w = 0.

    Therefore
    (v - w) = 0
    (w - u) = -1
    (v - u) = -1
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  6. #6
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    Quote Originally Posted by zorro View Post
    Is this correct ?

    a.b = u + v + w

    And since
    Mr F says: Equate components on each side. eg. Equating i components gives v - w = 0.

    Therefore
    (v - w) = 0
    (w - u) = -1
    (v - u) = -1
    Regarding a.b = u + v + w, recall what the question said that a.b was equal to. Therefore ....
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  7. #7
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    Is this correct?

    Quote Originally Posted by mr fantastic View Post
    Regarding a.b = u + v + w, recall what the question said that a.b was equal to. Therefore ....
    ok

     u + v + w = 1

    Mr fantastic is this the right step or no

    u = 1 - v - w

    Substituting u in (v - u) = -1 we get

    v - 1 + v + w = -1
    2v + w = 0
    v = \frac{-w}{2}

    Substituting v in (v - w) = 0 weget

    \frac{-w}{2} - w = 0

    \frac{-w - 2w}{2} = 0

    \frac{-3w}{2} = 0

    w = - \frac{2}{3}

    substituting w in (v - w) = 0

    v - \frac{2}{3} = 0

    v = \frac{2}{3}

    Now Subtitute v in (v - u) = 0

    \frac{2}{3} - u = 0

    u = \frac{2}{3}


    Am i doing it correctly or no ?????
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  8. #8
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    Quote Originally Posted by zorro View Post
    ok

     u + v + w = 1

    Mr fantastic is this the right step or no

    u = 1 - v - w

    Substituting u in (v - u) = -1 we get

    v - 1 + v + w = -1
    2v + w = 0
    v = \frac{-w}{2}

    Substituting v in (v - w) = 0 weget

    \frac{-w}{2} - w = 0

    \frac{-w - 2w}{2} = 0

    \frac{-3w}{2} = 0

    w = - \frac{2}{3}

    substituting w in (v - w) = 0

    v - \frac{2}{3} = 0

    v = \frac{2}{3}

    Now Subtitute v in (v - u) = 0

    \frac{2}{3} - u = 0

    u = \frac{2}{3}


    Am i doing it correctly or no ?????
    Have you checked your solution by substituting it into the equations?

    From the equations: v = w, u = 1 + v = 1 + w.

    Substitute into u + v + w = 1. Therefore ....
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  9. #9
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    Is this correct

    Quote Originally Posted by mr fantastic View Post
    Have you checked your solution by substituting it into the equations?

    From the equations: v = w, u = 1 + v = 1 + w.

    Substitute into u + v + w = 1. Therefore ....

    I am getting
    u = 1
    v = 0
    w = 0
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  10. #10
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    Quote Originally Posted by zorro View Post
    I am getting
    u = 1
    v = 0
    w = 0
    Correct.
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  11. #11
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    Thanks u every one for helping me

    Thank u mr fantastic and every one else for helping me
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