Find the shortest distance between the lines l1 and l2 whose vector equations are ` -> r= hat i+ hat j+lambda(2 hat i- hat j+ hat k)` (1)and ` -> r=2 hat i+ hat j-k+mu(3 hat i-5 hat j+2 hat k)` (2)

Explanation

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

### Step 1: Identify the lines and their parameters
The vector equations of the lines are given as:
1. Line \( l_1: \mathbf{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k}) \)
2. Line \( l_2: \mathbf{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(3\hat{i} - 5\hat{j} + 2\hat{k}) \)

From these equations, we can identify:
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