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Math Help - Cross Product proof?

  1. #1
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    Cross Product proof?

    I have to prove that (a - b) x (a + b) = 2(a x b)

    I get this:

    (a + -b) x c = 2(a x b)

    a x c + -b x c = 2(a x b)

    a x (a + b) + -b x (a + b)= 2(a x b)

    a x a + a x b + -b x a + -b x b= 2( a x b) - (a x b)
    -(a x b)

    0 + -b x a + -b x b = a x b

    a x b + -b x b = a x b - (a x b)
    -a x b

    -b x b = 0

    and I can't do anything

    Thanks for your help
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  2. #2
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    Quote Originally Posted by Warrenx View Post
    I have to prove that (a - b) x (a + b) = 2(a x b)

    I get this:

    (a + -b) x c = 2(a x b)

    a x c + -b x c = 2(a x b)

    a x (a + b) + -b x (a + b)= 2(a x b)

    a x a + a x b + -b x a + -b x b= 2( a x b) - (a x b)
    -(a x b)

    0 + -b x a + -b x b = a x b

    a x b + -b x b = a x b - (a x b)
    -a x b

    -b x b = 0

    and I can't do anything

    Thanks for your help
    \mathbf{a} = x_1\mathbf{i} + y_1\mathbf{j} + z_1\mathbf{k}


    \mathbf{b} = x_2\mathbf{i} + y_2\mathbf{j} + z_2\mathbf{k}.


    \mathbf{a} + \mathbf{b} = (x_1 + x_2)\mathbf{i} + (y_1 + y_2)\mathbf{j} + (z_1 + z_2)\mathbf{k}.


    \mathbf{a} - \mathbf{b} = (x_1 - x_2)\mathbf{i} + (y_1 - y_2)\mathbf{j} + (z_1 - z_2)\mathbf{k}.


    (\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b}) = \left|\begin{matrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ x_1 + x_2 & y_1 + y_2 & z_1 + z_2 \\ x_1 - x_2 & y_1 - y_2 & z_1 - z_2\end{matrix}\right|


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  3. #3
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    Quote Originally Posted by Warrenx View Post
    I have to prove that (a - b) x (a + b) = 2(a x b)

    I get this:

    (a + -b) x c = 2(a x b)

    a x c + -b x c = 2(a x b)

    a x (a + b) + -b x (a + b)= 2(a x b)

    a x a + a x b + -b x a + -b x b= 2( a x b) - (a x b)
    -(a x b)
    Are you saying that you are subtracting (a x b) from both sides?

    There is no need to do that. Yes, you have, on the left
    a x a + a x b - b x a+ b x b

    But the cross product is anti-symmetric: a x a= b x b= 0 and
    b x a= - a x b so
    a x a+ a x b- b x a + b x b= 0+ a x b- (-a x b)+ 0= 2 a x b.

    0 + -b x a + -b x b = a x b

    a x b + -b x b = a x b - (a x b)
    -a x b

    -b x b = 0

    and I can't do anything

    Thanks for your help
    You can't do anything because you went ahead and canceled everything and arrived at the trivially true statement "-b x b = 0"!
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