The moment of the couple formed by the forces $5\hat i+\hat k$ and $-5\hat i -\hat k$ acting at the points (9,-1, 2) and (3, -2, 1) respectively, is

$\hat i-\hat j-5\hat k$

We have,

Moment of the couple = $(\vec{r_1}-\vec{r_2})×\vec F$,

where $\vec{r_1}$ and $\vec{r_2}$ are the position vectors of the points of application of the forces $\vec F=5\hat i+\hat k$ and $-\vec F=-5\hat i-\hat k$ respectively.

Here, $\vec{r_1}-\vec{r_2} (9\hat i-\hat j + 2\hat k) -(3\hat i -2\hat j + \hat k) =6\hat i+\hat j+\hat k$

and, $\vec F=5\hat i+\hat k$

∴ Moment of the couple = $(\vec{r_1}-\vec{r_2}) × \vec F$

⇒ Moment of the couple = $(6\hat i+\hat j+\hat k)×(5\hat i+\hat k)=\hat i-\hat j-5\hat k$