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

4

Let $\vec a =\hat i+\hat j, \vec b =\hat j +\hat k, \vec c=\hat k+\hat i$.

Let $\vec{n_1},\vec{n_2},\vec{n_3}$ be the normals to the given planes. Then,

$\vec{n_1}=\vec a×\vec b,\vec{n_2}=\vec b×\vec c$ and $\vec{n_3}=\vec c×\vec a$

∴ Volume of the given parallelopiped is given by

$[\vec{n_1}\,\,\vec{n_2}\,\,\vec{n_3}]=[\vec a\,\,\vec b\,\,\vec c]^2=\begin{vmatrix}1&1&0\\0&1&1\\1&0&1\end{vmatrix}^2=4$