Find the shortest distance between the lines ` vec r=( hat i+2 hat j+ hat k)+lambda( hat i- hat j+ hat k)` and ` vec r=(2 hat i- hat j- hat k)+mu(2 hat i+ hat j+2 hat k)`

Explanation

To find the shortest distance between the two given lines, we can follow these steps:

### Step 1: Identify the lines and their components
The lines are given in vector form:
1. Line 1:
\[ \vec{r_1} = \hat{i} + 2\hat{j} + \hat{k} + \lambda(\hat{i} - \hat{j} + \hat{k}) \]
...