Find the shortest distance between the given line
`vec(r ) =(hat(i)+2hat(j) -4hat(k)) + lambda(2hat(i) +3hat(j) +6hat(k))`
`vec(r )=(3hat(i) +3hat(j) -5hat(k)) + mu (-2hat(i) +3hat(j) +8hat(k))`

Answer

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

### Step 1: Identify the lines
The two lines are given in vector form:
1. Line 1: \(\vec{r_1} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})\)
2. Line 2: \(\vec{r_2} = (3\hat{i} + 3\hat{j} - 5\hat{k}) + \mu(-2\hat{i} + 3\hat{j} + 8\hat{k})\)

### Step 2: Compare with the standard form
We can express these lines in the standard form:
- Line 1: \(\vec{r_1} = \vec{a_1} + \lambda \vec{b_1}\)
- Line 2: \(\vec{r_2} = \vec{a_2} + \mu \vec{b_2}\)

From the equations:
- \(\vec{a_1} = \hat{i} + 2\hat{j} - 4\hat{k}\)
- \(\vec{b_1} = 2\hat{i} + 3\hat{j} + 6\hat{k}\)
- \(\vec{a_2} = 3\hat{i} + 3\hat{j} - 5\hat{k}\)
- \(\vec{b_2} = -2\hat{i} + 3\hat{j} + 8\hat{k}\)

### Step 3: Find \(\vec{a_2} - \vec{a_1}\)
Now, we calculate:
\[ \vec{a_2} - \vec{a_1} = (3\hat{i} + 3\hat{j} - 5\hat{k}) - (\hat{i} + 2\hat{j} - 4\hat{k}) \]
This simplifies to:
\[ \vec{a_2} - \vec{a_1} = (3 - 1)\hat{i} + (3 - 2)\hat{j} + (-5 + 4)\hat{k} = 2\hat{i} + \hat{j} - \hat{k} \]

### Step 4: Calculate \(\vec{b_1} \times \vec{b_2}\)
Next, we find the cross product \(\vec{b_1} \times \vec{b_2}\):
\[ \vec{b_1} = \begin{pmatrix} 2 \\ 3 \\ 6 \end{pmatrix}, \quad \vec{b_2} = \begin{pmatrix} -2 \\ 3 \\ 8 \end{pmatrix} \]
Using the determinant:
\[ \vec{b_1} \times \vec{b_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 6 \\ -2 & 3 & 8 \end{vmatrix} \]
Calculating this determinant:
\[ = \hat{i}(3 \cdot 8 - 6 \cdot 3) - \hat{j}(2 \cdot 8 - 6 \cdot -2) + \hat{k}(2 \cdot 3 - 3 \cdot -2) \]
\[ = \hat{i}(24 - 18) - \hat{j}(16 + 12) + \hat{k}(6 + 6) \]
\[ = 6\hat{i} - 28\hat{j} + 12\hat{k} \]

### Step 5: Find the magnitude of \(\vec{b_1} \times \vec{b_2}\)
Now we calculate the magnitude:
\[ |\vec{b_1} \times \vec{b_2}| = \sqrt{6^2 + (-28)^2 + 12^2} \]
\[ = \sqrt{36 + 784 + 144} = \sqrt{964} \]
\[ = 2\sqrt{241} \]

### Step 6: Calculate the shortest distance
The formula for the shortest distance \(d\) between the two lines is:
\[ d = \frac{|(\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|} \]
Calculating the dot product:
\[ (\vec{a_2} - \vec{a_1}) = 2\hat{i} + \hat{j} - \hat{k} \]
\[ \vec{b_1} \times \vec{b_2} = 6\hat{i} - 28\hat{j} + 12\hat{k} \]
Calculating the dot product:
\[ (2\hat{i} + \hat{j} - \hat{k}) \cdot (6\hat{i} - 28\hat{j} + 12\hat{k}) = 2 \cdot 6 + 1 \cdot (-28) + (-1) \cdot 12 \]
\[ = 12 - 28 - 12 = -28 \]
Taking the absolute value:
\[ |(\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})| = 28 \]
Finally, substituting into the distance formula:
\[ d = \frac{28}{2\sqrt{241}} = \frac{14}{\sqrt{241}} \]

### Final Answer
The shortest distance between the given lines is:
\[ \frac{14}{\sqrt{241}} \]