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Math Help - Linear Algebra

  1. #1
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    Linear Algebra

    Hi there,

    I'm finding my algebra unit quite difficult compared to calculus... Could someone help with the below problem?

    Let m, n and o be three vectors in R3 such that
    m + n = o.
    (a) Show that m x n = o x n = m x o.
    (b) Use the geometric meaning of vector addition and cross
    product to explain the statement of (a) geometrically. Make
    sure that all the cases are covered.
    Thanks in advance!
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  2. #2
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    Re: Linear Algebra

    Let m=(m_1,m_2,m_3), n = (n_1,n_2,n_3), o = (o_1,o_2,o_3). Calculate each cross product. Since m+n=o, you know o = (m_1+n_1,m_2+n_2,m_3+n_3).
    Thanks from SydneyGuy
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  3. #3
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    Re: Linear Algebra

    You can also do this without considering coordinates. Using the fact that the cross product is linear in each argument and $a\times a=0$, multiply $m+n=o$ by $m$ on the left and by $n$ on the right.
    Thanks from SydneyGuy
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  4. #4
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    Re: Linear Algebra

    Thanks guys! Can anybody help with part b? I know the geometric meaning of cross product is |v x w| = |v||w|sin theta, but I'm not sure how to approach the problem
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  5. #5
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    Re: Linear Algebra

    I wouldn't consider that a "geometrical" meaning. Rather, the geometrical meaning of the cross product of two vectors is the area of the parallelogram having the two vectors as sides.
    Thanks from SydneyGuy
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  6. #6
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    Re: Linear Algebra

    It is also the vector that is normal (or perpendicular) to the plane containing the two vectors. The vector $m+n$ is in the same plane as both $m$ and $n$, so the same vector that is perpendicular to both $m$ and $n$ will also be perpendicular with $m$ and $m+n$ and $n$ and $m+n$.
    Thanks from SydneyGuy
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