The three vectors $\hat i+\hat j,\hat j+\hat k, \hat k+\hat i$ taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume

$\frac{4}{3\sqrt{3}}$

The correct answer is Option (4) → $\frac{4}{3\sqrt{3}}$

$(\hat i+\hat j)×(\hat j+\hat k)=\hat k-\hat j+\hat i$

$(\hat j+\hat k)×(\hat k+\hat i)=\hat i-\hat k+\hat j$

$(\hat k+\hat i)×(\hat i+\hat j)=\hat k-\hat i+\hat j$

Volume = $\frac{1}{3\sqrt{3}}\begin{bmatrix}\vec p&\vec q&\vec r\end{bmatrix}$

$=\frac{1}{3\sqrt{3}}\begin{vmatrix}1&-1&1\\1&1&1\\-1&1&1\end{vmatrix}$

$=\frac{1}{3\sqrt{3}}×4$