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

Explanation

To find the shortest distance between the two lines given by their vector equations, we will follow these steps:

### Step 1: Write the vector equations in the standard form
The vector equations of the lines are given as:
1. \( \vec{r_1} = (1-t) \hat{i} + (t-2) \hat{j} + (3-2t) \hat{k} \)
2. \( \vec{r_2} = (s+1) \hat{i} + (2s-1) \hat{j} + (-2s-1) \hat{k} \)

We can express these in the form \( \vec{r} = \vec{a} + \lambda \vec{b} \).
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