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Math Help - [SOLVED] show zero x vector in v3, (cross product)

  1. #1
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    [SOLVED] show zero x vector in v3, (cross product)

    must show that 0 x a and a x 0 = 0 for any vector a in V3

    so would this be sufficient:

    <0, 0, 0> x <1, 2, 3> = 0

    and <2, 3, 4> x <0, 0, 0> = 0

    because

    0 - 0 + 0 = 0

    correct?

    believe this is cross product stuff.
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  2. #2
    o_O
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    No. All you have shown is that this is true for a few vectors. You need to prove that it is true for all vectors.

    To show this, you let a be any general vector with components: a = \left< a_1, a_2, a_3\right>

    So let's take the cross product:
    0 \times a  = \left| \begin{matrix} i & j & k \\ 0 & 0 & 0 \\ a_1 & a_2 & a_3 \end{matrix} \right|  = \left| \begin{matrix} 0 & 0 \\ a_2 & a_3 \end{matrix} \right|i - \left| \begin{matrix} 0 & 0 \\ a_1 & a_3 \end{matrix} \right|j + \left| \begin{matrix} 0 & 0 \\ a_1 & a_2 \end{matrix} \right|j = \cdots

    Do the same with a \times 0.

    Both of these should be equal to 0 and since a was a general vector, then it must be true for any vector in \mathbb{R}^3.
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  3. #3
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    thanks, the other would be
    i, j, k
    |a1, a2, a3|
    |0, 0, 0|

    and the result 0 for both.

    i appreciate your help, thanks.
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