Vector products

Question

Vectors a, b and c are present.

Given $a+b+c=0$, show that $a\times b=b \times c= c\times a$

I have tried starting from $a+b=-c$ and $a+c=-b$

then $-b \times -c=(a+c)(a+b)$

But somehow its not making sense.

How does one prove it?

Answer 1

From $a\times (- a) =0$ and $-a=b+c$, one can get $$a\times (b+c)=0,$$ i.e., $a\times b=-a\times c=c\times a$. The other equality can be proved similarly.

Answer 2

$$ a+b+c=o \Rightarrow a+b=-c \Rightarrow (a+b)\times a=-c\times a \Rightarrow b\times a=a\times c \Rightarrow a\times b=c\times a $$ Can you figure out the rest ?