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Let: u, v, w \varepsilon R3

Prove: (u x v) * [(v x w) x (w x u)] = [u * (v x w)]2

User Iraklisg
by
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1 Answer

3 votes

Recall that for 3 vectors
a,b,c, all in
\mathbb R^3, the vector triple product


a*(b* c)=(a\cdot c)b-(a\cdot b)c

So


(v* w)*(w* u)=((v* w)\cdot u)w-((v* w)\cdot w)u

Also recall the scalar triple product,


a\cdot(b* c)

which gives the signed volume of the parallelipiped generated by the three vectors
a,b,c. When either
a=b or
a=c, the parallelipepid is degenerate and has 0 volume, so


(v* w)\cdot w=0

and the above reduces to


(v* w)*(w* u)=((v* w)\cdot u)w

so that


(u* v)\cdot[(v* w)*(w* u)]=(u* v)\cdot((v* w)\cdot u)w

The scalar triple product has the following property:


a\cdot(b* c)=b\cdot(c* a)=c\cdot(a* b)

Since
(v* w)\cdot u is a scalar, we can factor it out to get


((v* w)\cdot u)((u* v)\cdot w)

and by the property above we have


(u* v)\cdot w=u\cdot(v* w)

and so we end up with


[u\cdot(v* w)]^2

as required.

User Dahlia
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