A force of 39 kg. wt is acting at a point P(-4, 2, 5) in the direction $12\hat i-4\hat j-3\hat k$. The moment of this force about a line through the origin having the direction of $2\hat i-2\hat j+\hat k$, is

-76 units

We have,

Force ($\vec F$) = $39\frac{(12\hat i-4\hat j-3\hat k)}{\sqrt{144+16+9}}=36\hat i-12\hat j-9\hat k$

Let $\vec M$ be the moment of $\vec F$ about a point O (origin) on the given line. Then,

$\vec M=\vec r×vec F$, where $\vec r=\vec{OP}=-4\hat i+2\hat j+5\hat k$

$⇒\vec M=\begin{vmatrix}\hat i&\hat j&\hat k\\-4&2&5\\36&-12&-9\end{vmatrix}42\hat i+144\hat j-24\hat k$

So, the moment of $\vec F$ about a line having the direction $\vec a=2\hat i-2\hat j+\hat k$ is given by

$\vec M.\hat a=(42\hat i+144\hat j-24\hat k).\frac{2\hat i-2\hat j+\hat k}{\sqrt{4+4+1}}=-76$ units