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    Hi there. A bit unsure with a question.

    Use suffix notation to write  a \times (b \times c) - (a \times b) \times c as a simpler vector expression (involving no cross products).

    I'll put what I've done.

     a \times (b \times c) = e_i \epsilon_{ijk} a_j (b \times c)_k
     = e_i \epsilon_{ijk} a_j \epsilon_{klm} b_l c_m \\
     = e_i a_j b_l c_m (\delta_{il}\delta_{jm} - \delta_{im}\delta_{jl})
     = e_i a_j b_l c_m \delta_{il}\delta_{jm} - e_i a_j b_l c_m \delta_{im}\delta_{jl}

    And:

     (a \times b) \times c = e_i \epsilon_{ijk} c_j (a \times b)_k
     = ... = e_i c_j a_l b_m \delta_{il}\delta_{jm} - e_i c_j a_l b_m \delta_{im}\delta_{jl}

    So I get

     e_i a_m b_i c_m - e_i a_l b_l c_i

    for the first expression, and

     e_i a_i b_j c_j - e_i a_j b_i c_j (= e_i a_i b_j c_j - e_i a_m b_i c_m)

    for the second.

    So second subtracted from first gives

     e_i(2a_m b_i c_m - a_i b_j c_j - a_l b_l c_i)

    I can't help but feel the answer should be 0 (or something simpler). Can anyone see any mistakes?
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    (Original post by Goliathmo)
    Hi there. A bit unsure with a question.

    Use suffix notation to write  a \times (b \times c) - (a \times b) \times c as a simpler vector expression (involving no cross products).

    I'll put what I've done.

     a \times (b \times c) = e_i \epsilon_{ijk} a_j (b \times c)_k
     = e_i \epsilon_{ijk} a_j \epsilon_{klm} b_l c_m \\
     = e_i a_j b_l c_m (\delta_{il}\delta_{jm} - \delta_{im}\delta_{jl})
     = e_i a_j b_l c_m \delta_{il}\delta_{jm} - e_i a_j b_l c_m \delta_{im}\delta_{jl}

    And:

     (a \times b) \times c = e_i \epsilon_{ijk} c_j (a \times b)_k
     = ... = e_i c_j a_l b_m \delta_{il}\delta_{jm} - e_i c_j a_l b_m \delta_{im}\delta_{jl}

    So I get

     e_i a_m b_i c_m - e_i a_l b_l c_i

    for the first expression, and

     e_i a_i b_j c_j - e_i a_j b_i c_j (= e_i a_i b_j c_j - e_i a_m b_i c_m)

    for the second.

    So second subtracted from first gives

     e_i(2a_m b_i c_m - a_i b_j c_j - a_l b_l c_i)

    I can't help but feel the answer should be 0 (or something simpler). Can anyone see any mistakes?
    That last expression looks relatively simple to me. Try writing it out as a vector expression.
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    Looking only at your final answer, remember that using suffix notation, repeated indices are summed over. Think about a_m*c_m in the first term for instance.
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    (Original post by Slumpy)
    Looking only at your final answer, remember that using suffix notation, repeated indices are summed over. Think about a_m*c_m in the first term for instance.
    Ok. So we get?

     2b(a \cdot c) - a(b \cdot c) - c(a \cdot b)
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    That's correct.
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    (Original post by Slumpy)
    That's correct.
    Then I've made a mistake somewhere, because that isn't equivalent to the expression I was trying to simplify.
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    **** I think I know where I went wrong, I didn't consider the order of the vector product in the second expression.
 
 
 
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