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

Explanation

To find the shortest distance between the two lines given in vector form, we will follow these steps:

### Step 1: Identify the lines
The two lines are given as:
1. \( \vec{r_1} = \hat{i} + 2\hat{j} + \hat{k} + \lambda (\hat{i} - \hat{j} + \hat{k}) \)
2. \( \vec{r_2} = 2\hat{i} - \hat{j} - \hat{k} + \mu (2\hat{i} + \hat{j} + 2\hat{k}) \)

From these equations, we can identify:
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