Question

If $\left[\overline{a}\overline{b}\overline{c}\right]=1$ then $\frac{\overline{a}.(\overline{b}\times \overline{c})}{(\overline{c}\times \overline{a}).\overline{b}}+\frac{\overline{b}.(\overline{c}\times \overline{a})}{(\overline{b}\times \overline{a}).\overline{c}}+\frac{\overline{c}.(\overline{a}\times \overline{b})}{(\overline{b}\times \overline{c}).\overline{a}}=$

Answer 1

$\left[\overline{a}\overline{b}\overline{c}\right]=1$

$\Rightarrow \left[\overline{a}\overline{b}\overline{c}\right]=\left[\overline{b}\overline{c}\overline{a}\right]=\left[\overline{c}\overline{a}\overline{b}\right]=1$

$\Rightarrow \left[\overline{b}\overline{a}\overline{c}\right]=\left[\overline{a}\overline{c}\overline{b}\right]=\left[\overline{c}\overline{a}\overline{b}\right]=-1$

$\Rightarrow \frac{\overline{a}.(\overline{b}\times \overline{c})}{(\overline{c}\times \overline{a}).\overline{b}}+\frac{\overline{b}.(\overline{c}\times \overline{a})}{(\overline{b}\times \overline{a}).\overline{c}}+\frac{\overline{c}.(\overline{a}\times \overline{b})}{(\overline{b}\times \overline{c}).\overline{a}}$

$=\frac{\left[\overline{a}\overline{b}\overline{c}\right]}{\left[\overline{c}\overline{a}\overline{b}\right]}+\frac{\left[\overline{b}\overline{c}\overline{a}\right]}{\left[\overline{b}\overline{a}\overline{c}\right]}+\frac{\left[\overline{c}\overline{a}\overline{b}\right]}{\left[\overline{b}\overline{c}\overline{a}\right]}=\frac{1}{1}+\frac{1}{-1}+\frac{1}{1}=1$