# If $\overrightarrow {a}+ \overrightarrow {b}+ \overrightarrow {c}=\overrightarrow 0$ show that $\overrightarrow {a} \times \overrightarrow {b}=\overrightarrow {b} \times \overrightarrow {c}=\overrightarrow {c} \times \overrightarrow {a}$

Toolbox:
• $\large\overrightarrow a \times\overrightarrow b=-\overrightarrow b\times\overrightarrow a$
• $\large\overrightarrow a\times\overrightarrow a=\overrightarrow 0$
Given $\large\overrightarrow a+\overrightarrow b+\overrightarrow c=\overrightarrow 0$
$\large\Rightarrow\:\overrightarrow a+\overrightarrow b=-\overrightarrow c$
Taking cross product with $\large\overrightarrow c$ on both the sides we get
$\large(\overrightarrow a+\overrightarrow b)\times\overrightarrow c=-\overrightarrow c\times\overrightarrow c$
But we know that $\large\overrightarrow c\times\overrightarrow c=\overrightarrow 0$
$\Rightarrow\:(\large\overrightarrow a\times\overrightarrow c)+(\overrightarrow b\times\overrightarrow c)=\overrightarrow 0$
$\Rightarrow \:\large\overrightarrow b\times\overrightarrow c=-\overrightarrow a\times\overrightarrow c$
But we know that $\large-\overrightarrow a\times\overrightarrow c=\overrightarrow c\times\overrightarrow a$
$\Rightarrow \:\large\overrightarrow b\times\overrightarrow c=\overrightarrow c\times\overrightarrow a$...........(i)
Similarly from the given equation $\large\overrightarrow a+\overrightarrow b+\overrightarrow c=\overrightarrow 0$,
We can write $\large\overrightarrow b+\overrightarrow c=-\overrightarrow a$
Taking cross product with $\large\overrightarrow a$ on both sides we get
$\large(\overrightarrow b+\overrightarrow c)\times\overrightarrow a=-\overrightarrow a\times\overrightarrow a$
But we know that $\large\overrightarrow a\times\overrightarrow a=\overrightarrow 0$
$\Rightarrow\:(\large\overrightarrow b\times\overrightarrow a)+(\overrightarrow c\times\overrightarrow a)=\overrightarrow 0$
$\Rightarrow \:\large\overrightarrow c\times\overrightarrow a=-\overrightarrow b\times\overrightarrow a$
But we know that $\large-\overrightarrow b\times\overrightarrow a=\overrightarrow a\times\overrightarrow b$
$\Rightarrow \:\large\overrightarrow c\times\overrightarrow a=\overrightarrow a\times\overrightarrow b$...........(ii)
From (i) and (ii) we get the result
$\large\overrightarrow a\times\overrightarrow b=\overrightarrow b\times\overrightarrow c=\overrightarrow c\times\overrightarrow a$

edited Apr 17, 2013