Find the shortest distance between the lines ` vec r=(4 hat i- hat j)+lambda( hat i+2 hat j-3 hat k)a n d vec r=( hat i- hat j+2 hat k)+mu(2 hat i+4 hat j-5 hat k)dot` .

Explanation

To find the shortest distance between the two lines given by the equations:

1. \(\vec{r} = (4\hat{i} - \hat{j}) + \lambda(\hat{i} + 2\hat{j} - 3\hat{k})\)
2. \(\vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \mu(2\hat{i} + 4\hat{j} - 5\hat{k})\)

we will follow these steps:

### Step 1: Identify the vectors
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