Question

If $a,b$ and $c$ are three non-coplanar uni-modular vectors, each inclined with other at an angle 30${}^o$, then volume of tetrahedron whose edges are $a,b$ and $c$ is

• A. $\frac{\sqrt{3\sqrt{3}-5}}{12}$
• B. $\frac{3\sqrt{3}-5}{12}$
• C. $\frac{5\sqrt{2}+3}{12}$
• D. None of these

Answer 1

The correct option is A $\frac{\sqrt{3\sqrt{3}-5}}{12}$

Since, the volume of tetrahedron with edges $a,b$ and $c$ is $\left[abc\right]$.

Where, $a\cdot a=b\cdot b=c\cdot c=1$
and $a\cdot c=b\cdot c=c\cdot a=\frac{\sqrt{3}}{2}$ (given)
$\therefore V=\frac{1}{6}\left[abc\right]$
$\Rightarrow V^2=\frac{1}{36}[abc{]}^2=\frac{1}{36}\left|\begin{array}{ccc}a\cdot a& a\cdot b& a\cdot c\\ b\cdot a& b\cdot b& b\cdot c\\ c\cdot a& c\cdot b& c\cdot c\end{array}\right|$

$=\frac{1}{36}\left|\begin{array}{ccc}1& \frac{\sqrt{3}}{2}& \frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& 1& \frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& \frac{\sqrt{3}}{2}& 1\end{array}\right|=\frac{1}{36}(\frac{3\sqrt{3}}{4}-\frac{5}{4})$

$V=\frac{1}{12}\sqrt{3\sqrt{3}-5}$