Question

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $a,b,c$ such that $a.b=a.c=b.c=\frac{1}{2}.$ Then the volume of the parallelopiped is

• A. $\frac{1}{\sqrt{2}}$
• B. $\frac{1}{2\sqrt{2}}$
• C. $\frac{\sqrt{3}}{2}$
• D. $\frac{1}{\sqrt{3}}$

Answer 1

The correct option is A $\frac{1}{\sqrt{2}}$

$(\text{Volume}{)}^2=[abc\left]\right[abc]$
$=\left|\begin{array}{ccc}a\cdot a& b\cdot a& c\cdot c\\ a\cdot b& b\cdot b& c\cdot b\\ a\cdot c& b\cdot c& c\cdot c\end{array}\right|$
$=\left|\begin{array}{ccc}1& 1/2& 1/2\\ 1/2& 1& 1/2\\ 1/2& 1/2& 1\end{array}\right|=\frac{1}{2}$
$\therefore \text{Volume}=\frac{1}{\sqrt{2}}$