Browse Questions

# If $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are the sides $BC,CA\:and \:AB$ respectively of the $\Delta \:ABC$, then

$\begin{array}{1 1}\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow c+\overrightarrow c.\overrightarrow a=0 \\ \overrightarrow a\times\overrightarrow b=\overrightarrow b\times\overrightarrow c=\overrightarrow c\times\overrightarrow a \\ \overrightarrow a.\overrightarrow b=\overrightarrow b.\overrightarrow c=\overrightarrow c.\overrightarrow a \\ \overrightarrow a\times\overrightarrow b+\overrightarrow b\times\overrightarrow c+\overrightarrow c\times\overrightarrow a=0 \end{array}$

Toolbox:
• In any $\Delta\:ABC$, $\overrightarrow {BC}+\overrightarrow {CA}+\overrightarrow {AB}=0$
In any $\Delta\:ABC$, $\overrightarrow {BC}+\overrightarrow {CA}+\overrightarrow {AB}=0$
$\Rightarrow\:\overrightarrow a+\overrightarrow b+\overrightarrow c=0$
$\Rightarrow\:\overrightarrow a+\overrightarrow b=-\overrightarrow c$
Taking cross product with $\overrightarrow c$ on both the sides
$\overrightarrow c\times (\overrightarrow a+\overrightarrow b)=-\overrightarrow c\times\overrightarrow c=0$
$\Rightarrow\:\overrightarrow c\times\overrightarrow a+\overrightarrow c\times\overrightarrow b=0$
$\Rightarrow\:\overrightarrow c\times\overrightarrow a=\overrightarrow b\times\overrightarrow c=0$............(i)
Similarly by writing $\overrightarrow b+\overrightarrow c=-\overrightarrow a$ and by taking cross product with $\overrightarrow a$ on both sides,
we get $\overrightarrow a\times\overrightarrow b=\overrightarrow c\times\overrightarrow a=0$............(ii)
From (i) and (ii) we get $\overrightarrow a\times\overrightarrow b=\overrightarrow b\times\overrightarrow c=\overrightarrow c\times\overrightarrow a$