If d is the shortest distance between the lines `vecr =(3hati +5hatj + 7hatk)+lambda (hati +2hatj +hatk)` and
`vecr = (-hati -hatj-hatk)+mu(7hati-6hatj+hatk)` then `125d^(2)` is equal to ____________.

Answer

To solve the problem, we need to find the shortest distance \( d \) between the two given lines and then compute \( 125d^2 \).

### Step 1: Identify the lines and their parameters

The first line is given by:
\[ \vec{r_1} = (3\hat{i} + 5\hat{j} + 7\hat{k}) + \lambda (\hat{i} + 2\hat{j} + \hat{k}) \]
This means:
- Point on line 1, \( \vec{a_1} = 3\hat{i} + 5\hat{j} + 7\hat{k} \)
- Direction vector of line 1, \( \vec{b_1} = \hat{i} + 2\hat{j} + \hat{k} \)

The second line is given by:
\[ \vec{r_2} = (-\hat{i} - \hat{j} - \hat{k}) + \mu (7\hat{i} - 6\hat{j} + \hat{k}) \]
This means:
- Point on line 2, \( \vec{a_2} = -\hat{i} - \hat{j} - \hat{k} \)
- Direction vector of line 2, \( \vec{b_2} = 7\hat{i} - 6\hat{j} + \hat{k} \)

### Step 2: Check if the lines are parallel, intersecting, or skew

To check if the lines are parallel, we need to see if the direction vectors \( \vec{b_1} \) and \( \vec{b_2} \) are proportional. We can do this by checking if there exists a scalar \( k \) such that:
\[ \vec{b_1} = k \vec{b_2} \]
Calculating the cross product \( \vec{b_1} \times \vec{b_2} \) will help us determine if they are parallel.

### Step 3: Calculate the cross product \( \vec{b_1} \times \vec{b_2} \)

\[ \vec{b_1} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \quad \vec{b_2} = \begin{pmatrix} 7 \\ -6 \\ 1 \end{pmatrix} \]
The cross product is calculated using the determinant:
\[ \vec{b_1} \times \vec{b_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 1 \\ 7 & -6 & 1 \end{vmatrix} \]
Calculating this determinant:
\[ = \hat{i}(2 \cdot 1 - 1 \cdot (-6)) - \hat{j}(1 \cdot 1 - 1 \cdot 7) + \hat{k}(1 \cdot (-6) - 2 \cdot 7) \]
\[ = \hat{i}(2 + 6) - \hat{j}(1 - 7) + \hat{k}(-6 - 14) \]
\[ = 8\hat{i} + 6\hat{j} - 20\hat{k} \]

### Step 4: Calculate the magnitude of the cross product

The magnitude of \( \vec{b_1} \times \vec{b_2} \) is:
\[ |\vec{b_1} \times \vec{b_2}| = \sqrt{8^2 + 6^2 + (-20)^2} = \sqrt{64 + 36 + 400} = \sqrt{500} = 5\sqrt{10} \]

### Step 5: Calculate \( \vec{a_2} - \vec{a_1} \)

\[ \vec{a_2} - \vec{a_1} = (-\hat{i} - \hat{j} - \hat{k}) - (3\hat{i} + 5\hat{j} + 7\hat{k}) = (-4\hat{i} - 6\hat{j} - 8\hat{k}) \]

### Step 6: Calculate the dot product \( (\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1}) \)

\[ (\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1}) = (8\hat{i} + 6\hat{j} - 20\hat{k}) \cdot (-4\hat{i} - 6\hat{j} - 8\hat{k}) \]
Calculating the dot product:
\[ = 8(-4) + 6(-6) + (-20)(-8) = -32 - 36 + 160 = 92 \]

### Step 7: Calculate the shortest distance \( d \)

Using the formula for the shortest distance:
\[ d = \frac{|(\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1})|}{|\vec{b_1} \times \vec{b_2}|} = \frac{92}{5\sqrt{10}} \]

### Step 8: Calculate \( 125d^2 \)

First, calculate \( d^2 \):
\[ d^2 = \left(\frac{92}{5\sqrt{10}}\right)^2 = \frac{8464}{250} = \frac{8464}{250} \]

Now, calculate \( 125d^2 \):
\[ 125d^2 = 125 \cdot \frac{8464}{250} = \frac{125 \cdot 8464}{250} = \frac{1058000}{250} = 2116 \]

Thus, the final answer is:
\[ \boxed{2116} \]