Find the value of k so that the lines `(1-x)/(3) = (y-2)/(2k) = (z-3)/(2)` and `(1+x)/(3k) = (y-1)/(1) = (6-z)/(7)` are perpendicular to each other.

Answer

To find the value of \( k \) such that the given lines are perpendicular to each other, we will follow these steps:

### Step 1: Identify the direction ratios of the lines

The first line is given by:
\[ \frac{1-x}{3} = \frac{y-2}{2k} = \frac{z-3}{2} \]
From this, we can extract the direction ratios:
- For \( x \): \( -3 \)
- For \( y \): \( 2k \)
- For \( z \): \( 2 \)

Thus, the direction ratios of the first line are \( a_1 = -3 \), \( b_1 = 2k \), \( c_1 = 2 \).

The second line is given by:
\[ \frac{1+x}{3k} = \frac{y-1}{1} = \frac{6-z}{7} \]
From this, we can extract the direction ratios:
- For \( x \): \( 3k \)
- For \( y \): \( 1 \)
- For \( z \): \( -7 \)

Thus, the direction ratios of the second line are \( a_2 = 3k \), \( b_2 = 1 \), \( c_2 = -7 \).

### Step 2: Use the condition for perpendicularity

For two lines to be perpendicular, the following condition must hold:
\[ a_1 \cdot a_2 + b_1 \cdot b_2 + c_1 \cdot c_2 = 0 \]

Substituting the values we found:
\[ (-3)(3k) + (2k)(1) + (2)(-7) = 0 \]

### Step 3: Simplify the equation

Now, simplifying the equation:
\[ -9k + 2k - 14 = 0 \]
Combine like terms:
\[ -7k - 14 = 0 \]

### Step 4: Solve for \( k \)

Rearranging gives:
\[ -7k = 14 \]
Dividing both sides by -7:
\[ k = -2 \]

### Conclusion

Thus, the value of \( k \) such that the lines are perpendicular to each other is:
\[ \boxed{-2} \]