If the lines `(x-1)/(-3) =(y-2)/(2lambda) =(z-3)/(2) " and " (x-1)/(3lambda) =(y-1)/(1)=(6-z)/(5)` are perpendicular to each other then find the value of `lambda`

Answer

To solve the problem, we need to find the value of \( \lambda \) such that the two given lines are perpendicular to each other.

### Step-by-Step Solution:

1. **Identify the equations of the lines:**
The first line is given by:
\[ \frac{x-1}{-3} = \frac{y-2}{2\lambda} = \frac{z-3}{2} \]
The second line is given by:
\[ \frac{x-1}{3\lambda} = \frac{y-1}{1} = \frac{6-z}{5} \]

2. **Extract the direction ratios:**
For the first line, the direction ratios can be extracted from the denominators:
- Direction ratios of line 1, \( l_1 \): \( (-3, 2\lambda, 2) \)

For the second line, we rewrite the equation for clarity:
\[ \frac{x-1}{3\lambda} = \frac{y-1}{1} = \frac{z-6}{-5} \]
Thus, the direction ratios are:
- Direction ratios of line 2, \( l_2 \): \( (3\lambda, 1, -5) \)

3. **Use the condition for perpendicular lines:**
Two lines are perpendicular if the dot product of their direction ratios is zero. Therefore, we set up the equation:
\[ (-3)(3\lambda) + (2\lambda)(1) + (2)(-5) = 0 \]

4. **Simplify the equation:**
Expanding the equation gives:
\[ -9\lambda + 2\lambda - 10 = 0 \]
Combining like terms results in:
\[ -7\lambda - 10 = 0 \]

5. **Solve for \( \lambda \):**
Rearranging the equation gives:
\[ -7\lambda = 10 \]
Dividing both sides by -7 results in:
\[ \lambda = -\frac{10}{7} \]

### Final Answer:
\[ \lambda = -\frac{10}{7} \]