Find the shortest distance between the lines ` -> r=( hat i+2 hat j+ hat k)+lambda( hat i- hat j+ hat k)`and ` -> 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 will follow these steps:

### Step 1: Identify the equations of the lines
The equations of the lines are given as:
1. Line 1: \( \vec{r} = \hat{i} + 2\hat{j} + \hat{k} + \lambda(\hat{i} - \hat{j} + \hat{k}) \)
2. Line 2: \( \vec{r} = 2\hat{i} - \hat{j} - \hat{k} + \mu(2\hat{i} + \hat{j} + 2\hat{k}) \)

### Step 2: Extract the direction vectors and points
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