The moment of the couple consisting of the force $\vec F=3\hat i+2\hat j-\hat k$ acting through the point $\hat i-\hat j+\hat k$ and $-\vec F$ acting through the point $2\hat i-3\hat j-\hat k$, is

$5\sqrt{5}$

Let A and B be the points having position vectors $\hat i-\hat j+\hat k$ and $2\hat i -3\hat j-k$ respectively. Then,

$\vec r=\vec{BA}=(\hat i -\hat j +\hat k) - (2\hat i - 3\hat j −\hat k) = -\hat i + 2\hat j + 2\hat k$

Let $\vec M$ be the vector moment of the couple. Then,

$\vec M=\vec r×\vec F\begin{vmatrix}\hat i&\hat j&\hat k\\-1&2&2\\3&2&-1\end{vmatrix}=-6\hat i+5\hat j-8\hat k$

∴ Moment of the couple = $|\vec M|=\sqrt{36 + 25+ 64} =5\sqrt{5}$