Let the line of the shortest distance between the lines
`L_1 : vec r = (hat i + 2 hat j + 3 hat k) + lambda (hat i - hat j + hat k)` and `L_2 : vec r = (4 hat i + 5 hat j + 6 hat k) + mu (hat i + hat j - hat k)`
intersect `L_1` and `L_2` at P and Q respectively. If `(alpha, beta, gamma)` is the mid point of the line segment PQ, then `2(alpha + beta + gamma)` is equal to ___________.

Answer

To solve the problem, we need to find the mid-point of the shortest distance between the lines \( L_1 \) and \( L_2 \) and then calculate \( 2(\alpha + \beta + \gamma) \).

### Step 1: Write the equations of the lines in vector form.
The lines are given as:
- \( L_1: \vec{r} = (1 \hat{i} + 2 \hat{j} + 3 \hat{k}) + \lambda (\hat{i} - \hat{j} + \hat{k}) \)
- \( L_2: \vec{r} = (4 \hat{i} + 5 \hat{j} + 6 \hat{k}) + \mu (\hat{i} + \hat{j} - \hat{k}) \)

### Step 2: Express the points on the lines.
The points on the lines can be expressed as:
- For \( L_1 \):
\[ P = (1 + \lambda) \hat{i} + (2 - \lambda) \hat{j} + (3 + \lambda) \hat{k} \]
- For \( L_2 \):
\[ Q = (4 + \mu) \hat{i} + (5 + \mu) \hat{j} + (6 - \mu) \hat{k} \]

### Step 3: Find the vector \( \vec{PQ} \).
The vector \( \vec{PQ} \) is given by:
\[ \vec{PQ} = Q - P = \left[(4 + \mu) - (1 + \lambda)\right] \hat{i} + \left[(5 + \mu) - (2 - \lambda)\right] \hat{j} + \left[(6 - \mu) - (3 + \lambda)\right] \hat{k} \]
This simplifies to:
\[ \vec{PQ} = (3 + \mu - \lambda) \hat{i} + (3 + \mu + \lambda) \hat{j} + (3 - \mu - \lambda) \hat{k} \]

### Step 4: Set up the conditions for perpendicularity.
Since \( \vec{PQ} \) is the shortest distance, it must be perpendicular to both direction vectors of \( L_1 \) and \( L_2 \).

1. Direction vector of \( L_1 \): \( \hat{b_1} = (1, -1, 1) \)
2. Direction vector of \( L_2 \): \( \hat{b_2} = (1, 1, -1) \)

Set up the dot products:
1. \( \vec{PQ} \cdot \hat{b_1} = 0 \):
\[ (3 + \mu - \lambda) - (3 + \mu + \lambda) + (3 - \mu - \lambda) = 0 \]
Simplifying gives:
\[ 3 - 2\lambda = 0 \implies \lambda = \frac{3}{2} \]

2. \( \vec{PQ} \cdot \hat{b_2} = 0 \):
\[ (3 + \mu - \lambda) + (3 + \mu + \lambda) - (3 - \mu - \lambda) = 0 \]
Simplifying gives:
\[ 3 + 3\mu = 0 \implies \mu = -1 \]

### Step 5: Find the coordinates of points \( P \) and \( Q \).
Substituting \( \lambda = \frac{3}{2} \) into \( P \):
\[ P = \left(1 + \frac{3}{2}\right) \hat{i} + \left(2 - \frac{3}{2}\right) \hat{j} + \left(3 + \frac{3}{2}\right) \hat{k} = \frac{5}{2} \hat{i} + \frac{1}{2} \hat{j} + \frac{9}{2} \hat{k} \]

Substituting \( \mu = -1 \) into \( Q \):
\[ Q = (4 - 1) \hat{i} + (5 - 1) \hat{j} + (6 + 1) \hat{k} = 3 \hat{i} + 4 \hat{j} + 7 \hat{k} \]

### Step 6: Find the midpoint \( M \) of segment \( PQ \).
The midpoint \( M \) is given by:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) \]
Calculating:
\[ \alpha = \frac{\frac{5}{2} + 3}{2} = \frac{11}{4}, \quad \beta = \frac{\frac{1}{2} + 4}{2} = \frac{9}{4}, \quad \gamma = \frac{\frac{9}{2} + 7}{2} = \frac{23}{4} \]

### Step 7: Calculate \( 2(\alpha + \beta + \gamma) \).
\[ \alpha + \beta + \gamma = \frac{11}{4} + \frac{9}{4} + \frac{23}{4} = \frac{43}{4} \]
Thus,
\[ 2(\alpha + \beta + \gamma) = 2 \cdot \frac{43}{4} = \frac{86}{4} = 21.5 \]

### Final Answer:
The value of \( 2(\alpha + \beta + \gamma) \) is \( 21.5 \).