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 equations:

1. \(\vec{r_1} = (1\hat{i} + 2\hat{j} + 1\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})\)
2. \(\vec{r_2} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k})\)

we can follow these steps:

### Step 1: Identify the Direction Vectors and Points on Each Line
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