The shortest distance between the line `L_1=(hat i - hat j + hat k) + lambda (2 hat i - 14 hat j + 5 hat k)` and `L_2=(hat j + hat k) + mu (-2 hat i - 4 hat + 7 hat)` then `L_1` and `L_2` is

Options

A. `frac{5}{sqrt (221)}`
B. `frac{10}{sqrt (221)}`
C. `frac{2}{sqrt (221)}`
D. `frac{5}{11}`

Answer

To find the shortest distance between the two lines \( L_1 \) and \( L_2 \), we will follow these steps:

### Step 1: Identify the lines and their vector representations
The lines are given as:
- \( L_1: \mathbf{r_1} = (\hat{i} - \hat{j} + \hat{k}) + \lambda (2\hat{i} - 14\hat{j} + 5\hat{k}) \)
- \( L_2: \mathbf{r_2} = (\hat{j} + \hat{k}) + \mu (-2\hat{i} - 4\hat{j} + 7\hat{k}) \)

From this, we can identify:
- Point on \( L_1 \) (let's call it \( \mathbf{a} \)): \( (1, -1, 1) \)
- Direction vector of \( L_1 \) (let's call it \( \mathbf{p} \)): \( (2, -14, 5) \)
- Point on \( L_2 \) (let's call it \( \mathbf{b} \)): \( (0, 1, 1) \)
- Direction vector of \( L_2 \) (let's call it \( \mathbf{q} \)): \( (-2, -4, 7) \)

### Step 2: Calculate the cross product of the direction vectors
We need to calculate \( \mathbf{p} \times \mathbf{q} \):
\[ \mathbf{p} = (2, -14, 5), \quad \mathbf{q} = (-2, -4, 7) \]
Using the determinant to find the cross product:
\[ \mathbf{p} \times \mathbf{q} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -14 & 5 \\ -2 & -4 & 7 \end{vmatrix} \]
Calculating this determinant:
\[ = \hat{i}((-14)(7) - (5)(-4)) - \hat{j}((2)(7) - (5)(-2)) + \hat{k}((2)(-4) - (-14)(-2)) \]
\[ = \hat{i}(-98 + 20) - \hat{j}(14 + 10) + \hat{k}(-8 - 28) \]
\[ = \hat{i}(-78) - \hat{j}(24) - \hat{k}(36) \]
Thus,
\[ \mathbf{p} \times \mathbf{q} = (-78, -24, -36) \]

### Step 3: Calculate \( \mathbf{b} - \mathbf{a} \)
Now we calculate \( \mathbf{b} - \mathbf{a} \):
\[ \mathbf{b} - \mathbf{a} = (0, 1, 1) - (1, -1, 1) = (-1, 2, 0) \]

### Step 4: Calculate the dot product \( (\mathbf{b} - \mathbf{a}) \cdot (\mathbf{p} \times \mathbf{q}) \)
Now we find the dot product:
\[ (\mathbf{b} - \mathbf{a}) \cdot (\mathbf{p} \times \mathbf{q}) = (-1, 2, 0) \cdot (-78, -24, -36) \]
Calculating this gives:
\[ = (-1)(-78) + (2)(-24) + (0)(-36) = 78 - 48 + 0 = 30 \]

### Step 5: Calculate the magnitude of \( \mathbf{p} \times \mathbf{q} \)
Now we calculate the magnitude of \( \mathbf{p} \times \mathbf{q} \):
\[ |\mathbf{p} \times \mathbf{q}| = \sqrt{(-78)^2 + (-24)^2 + (-36)^2} \]
Calculating this gives:
\[ = \sqrt{6084 + 576 + 1296} = \sqrt{7956} \]

### Step 6: Calculate the shortest distance
The formula for the shortest distance \( d \) between two skew lines is given by:
\[ d = \frac{|(\mathbf{b} - \mathbf{a}) \cdot (\mathbf{p} \times \mathbf{q})|}{|\mathbf{p} \times \mathbf{q}|} \]
Substituting the values we calculated:
\[ d = \frac{|30|}{\sqrt{7956}} \]
Calculating \( \sqrt{7956} \approx 89.2 \) gives:
\[ d \approx \frac{30}{89.2} \approx 0.336 \]

### Final Result
Thus, the shortest distance between the lines \( L_1 \) and \( L_2 \) is approximately \( 0.336 \).