Question

Show that for any three vector a, b, c
a*(b*c)+b*(c*a) +c*(a*b) =0

Answer 1

We have:

$$\bold{a\times(b\times c)=b(a\cdot c)-c(a\cdot b)}$$

$$\bold{b\times(c\times a)=c(b\cdot a)-a(b\cdot c)}$$

$$\bold{c\times(a\times b)=a(c\cdot b)-b(c\cdot a)}$$

And

$$\bold{(a\cdot c)=(c\cdot a), (a\cdot b)=(b\cdot a), (b\cdot c)=(c\cdot b)}$$

Thus,

$$\bold{a\times(b\times c)+b\times(c\times a)+c\times(a\times b)=0}$$