Find the shortest distance between the lines whose vector equations are ` -> r=( hat i+2 hat j+3 hat k)+lambda( hat i-3 hat j+2 hat k)`and ` -> r=4 hat i+5 hat j+6 hat k+mu(2 hat i+3 hat j+ hat k)`.

Explanation

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

### Step 1: Identify the direction vectors and points on the lines
The vector equations of the lines are given as:
1. \( \vec{r_1} = \hat{i} + 2\hat{j} + 3\hat{k} + \lambda(\hat{i} - 3\hat{j} + 2\hat{k}) \)
2. \( \vec{r_2} = 4\hat{i} + 5\hat{j} + 6\hat{k} + \mu(2\hat{i} + 3\hat{j} + \hat{k}) \)

From these equations, we can identify:
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