1. ## Vectors

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

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

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

Just using definitions, I get $|\bold{a}||\bold{a} \times \bold{b}| \cos \theta = |\bold{a}| ab \sin \theta \cos \theta = a^{2}b \sin \theta \cos \theta$

2. Originally Posted by heathrowjohnny
1. Prove that the cross product is non-associative.

So basically I want to prove that $(\bold{r} \times \bold{s}) \times \bold{t} \neq \bold{r} \times (\bold{s} \times \bold{t})$. Now I probably just need to find a counterexample right? So I can just use $\bold{i}, \bold{j}$ and $\bold{k}$? 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 $\bold{a} \cdot \bold{a} \times \bold{b} = 0$

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