Find the shortest distance between the lines whose vector equations are ` vec r=(1-t) hat i+(t-2) hat j+(3-2t) hat k` and ` vec r=(s+1) hat i+(2s-1) hat j-(2s+1) hat k`

Explanation

To find the shortest distance between the given lines, we will follow a systematic approach to express the lines in standard form, identify the direction vectors, and then apply the formula for the shortest distance between skew lines.

### Step-by-Step Solution:

1. **Identify the vector equations of the lines:**
The first line is given by:
\[
\vec{r_1} = (1-t) \hat{i} + (t-2) \hat{j} + (3-2t) \hat{k}
...