1. Let the given points be

$A(1,-1,3), \quad B(2,-4,11)$

and

$C(-1,2,3), \quad D(3,-2,10).$

We need the projection of the line segment $AB$ on the line $CD$.

2. Find the direction vectors:

For $AB$, $\vec{AB}=B-A=(2-1,\,-4-(-1),\,11-3)=(1,-3,8).$

For $CD$, $\vec{CD}=D-C=(3-(-1),\,-2-2,\,10-3)=(4,-4,7).$

3. The projection of vector $\vec{AB}$ on the line in the direction of $\vec{CD}$ is the scalar projection:

$\text{Projection} = \frac{\vec{AB}\cdot \vec{CD}}{|\vec{CD}|}.$

4. Compute the dot product:

$\vec{AB}\cdot \vec{CD} = (1)(4)+(-3)(-4)+(8)(7)=4+12+56=72.$

5. Compute the magnitude of $\vec{CD}$:

$|\vec{CD}|=\sqrt{4^2+(-4)^2+7^2} = \sqrt{16+16+49} = \sqrt{81}=9.$

6. Therefore,

$\text{Projection} = \frac{72}{9}=8.$

So, the required projection is

$\boxed{8}.$