Is this equation possible?

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Answer 1 · 2018-01-20T10:01:46

See below.

Calling

#vec u = (a,b,c)# and
#vec v = (b,c,a)#

we have

#norm(vec u)^2 ge << vec u, vec v >># then

#norm(vec u)^2/(<< vec u, vec v >>) ge 1#

concluding the equation is possible only when #a=b=c# non null

Answer 2 · 2018-01-20T10:25:06

#sintheta=(a^2+b^2+c^2)/(ab+bc+ca)#

#=>1/sintheta=(ab+bc+ca)/(a^2+b^2+c^2)#

#=>1-1/sintheta=1-(ab+bc+ca)/(a^2+b^2+c^2)#

#=>1-1/sintheta=((a^2+b^2+c^2)-(ab+bc+ca))/(a^2+b^2+c^2)#

#=>1-1/sintheta=(2(a^2+b^2+c^2)-2(ab+bc+ca))/(2(a^2+b^2+c^2))#

#=>1-1/sintheta=((a-b)^2+(b-c)^2+(c-a)^2)/(2(a^2+b^2+c^2))#

for real values of #a,b,c->" " RHS>=0#

#=>1-1/sintheta>=0#

#=>1/sintheta<=1#

#=>sintheta>=1#

But

#sintheta>1 " is not possible" #

#sintheta=1 " is possible" #

And so relation is valid for #sintheta=1# when we have #a=b=c#