Answer to Question #96652 in Analytic Geometry for Manisha Nayak

Question #96652
Show that for any three vector a, b, c
a*(b*c)+b*(c*a) +c*(a*b) =0
1
Expert's answer
2019-10-17T11:24:46-0400

We have:

a×(b×c)=b(ac)c(ab)\bold{a\times(b\times c)=b(a\cdot c)-c(a\cdot b)}

b×(c×a)=c(ba)a(bc)\bold{b\times(c\times a)=c(b\cdot a)-a(b\cdot c)}

c×(a×b)=a(cb)b(ca)\bold{c\times(a\times b)=a(c\cdot b)-b(c\cdot a)}

And


(ac)=(ca),(ab)=(ba),(bc)=(cb)\bold{(a\cdot c)=(c\cdot a), (a\cdot b)=(b\cdot a), (b\cdot c)=(c\cdot b)}

Thus,


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


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