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 lines given by the vector equations:

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 1: Identify Points and Direction Vectors
From the equations, we can identify:
- For Line 1:
...