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Thread: Vectors

  1. #1
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    Vectors

    1. Prove that the cross product is non-associative.

    So basically I want to prove that $\displaystyle (\bold{r} \times \bold{s}) \times \bold{t} \neq \bold{r} \times (\bold{s} \times \bold{t}) $. Now I probably just need to find a counterexample right? So I can just use $\displaystyle \bold{i}, \bold{j} $ and $\displaystyle \bold{k} $?


    2. Prove that $\displaystyle \bold{a} \cdot \bold{a} \times \bold{b} = 0 $

    Just using definitions, I get $\displaystyle |\bold{a}||\bold{a} \times \bold{b}| \cos \theta = |\bold{a}| ab \sin \theta \cos \theta = a^{2}b \sin \theta \cos \theta$
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  2. #2
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    Quote Originally Posted by heathrowjohnny View Post
    1. Prove that the cross product is non-associative.

    So basically I want to prove that $\displaystyle (\bold{r} \times \bold{s}) \times \bold{t} \neq \bold{r} \times (\bold{s} \times \bold{t}) $. Now I probably just need to find a counterexample right? So I can just use $\displaystyle \bold{i}, \bold{j} $ and $\displaystyle \bold{k} $? Mr F says: Yep. But I have a feeling that proof by counter-example wasn't the sort of proof the folks who set the question had in mind .....

    2. Prove that $\displaystyle \bold{a} \cdot \bold{a} \times \bold{b} = 0 $

    Just using definitions, I get $\displaystyle |\bold{a}||\bold{a} \times \bold{b}| \cos \theta = |\bold{a}| ab \sin \theta \cos \theta = a^{2}b \sin \theta \cos \theta$ Mr F says: $\displaystyle \bold{a} \times \bold{b}$ gives a vector perpendicular to $\displaystyle \bold{a}$ and $\displaystyle \bold{b}$ ...... So the dot product of this vector with $\displaystyle \bold{a}$ will obviously be zero since it's perpendicular to $\displaystyle \bold{a}$...
    ..
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