$\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}\text{ and }\overrightarrow{\mathrm{c}}$are three non-zero & non-coplanar vectors with fixed magnitudes. Angles between any of the vector with the normal of the plane containing other two is $\mathrm{\alpha }$. Volume of parallelepiped formed is V and its surface area is S. Which of the following is/are true?

$\mathrm{V}=|\overrightarrow{\mathrm{a}}\cdot (\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}\left)\right|=\left|\overrightarrow{\mathrm{a}}\right||\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}||\cos \mathrm{\alpha }|$

Similarly,$\mathrm{V}=\left|\overrightarrow{\mathrm{b}}\right||\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}}||\cos \mathrm{\alpha }|$

& $\mathrm{V}=\left|\mathrm{c}\right||\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}||\cos \mathrm{\alpha }|$

While$\mathrm{S}=2\left(\right|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}|+|\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}}\left|\right)$

$=2\left(\frac{\mathrm{V}}{\left|\overrightarrow{\mathrm{c}}\right||\cos \mathrm{\alpha }|}+\frac{\mathrm{V}}{\left|\overrightarrow{\mathrm{a}}\right||\cos \mathrm{\alpha }|}+\frac{\mathrm{V}}{\left|\overrightarrow{\mathrm{b}}\right||\cos \mathrm{\alpha }|}\right)$

$\Rightarrow \frac{\mathrm{V}}{\mathrm{S}}=\frac{|\cos \mathrm{\alpha }|}{2}{\left[\frac{1}{\left|\overrightarrow{\mathrm{a}}\right|}+\frac{1}{\left|\overrightarrow{\mathrm{b}}\right|}+\frac{1}{\left|\overrightarrow{\mathrm{c}}\right|}\right]}^{-1}$

Now,  $\mathrm{V}=|\overrightarrow{\mathrm{a}}\cdot (\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}\left)\right|=\left|\overrightarrow{\mathrm{a}}\right||\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}||\cos \mathrm{\alpha }|$

$=\left|\overrightarrow{\mathrm{a}}\right|\left|\overrightarrow{\mathrm{b}}\right|\left|\overrightarrow{\mathrm{c}}\right|\sin (\overrightarrow{\mathrm{b}},\overrightarrow{\mathrm{c}})|\cos \alpha |\le \overrightarrow{\mathrm{a}}\left|\right|\overrightarrow{\mathrm{b}}\left|\right|\overrightarrow{\mathrm{c}}\mid$

Also, $\sin (\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}})=\sin (\overrightarrow{\mathrm{b}},\overrightarrow{\mathrm{c}})=\sin (\overrightarrow{\mathrm{c}},\overrightarrow{\mathrm{a}})$

So, it not necessary that they are equally inclined.