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Archy [21]
3 years ago
8

Let: u, v, w \varepsilon R3 Prove: (u x v) * [(v x w) x (w x u)] = [u * (v x w)]2

Mathematics
1 answer:
sweet [91]3 years ago
4 0

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

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

So

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

Also recall the scalar triple product,

a\cdot(b\times 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\times w)\cdot w=0

and the above reduces to

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

so that

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

The scalar triple product has the following property:

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

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

((v\times w)\cdot u)((u\times v)\cdot w)

and by the property above we have

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

and so we end up with

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

as required.

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