If the points A( 5,5 `lamda`) , B ( - 1, 3 , 2) and C ( -4 , 2 , -2) are collinear , find the value of `lamda` .

Answer

To find the value of \( \lambda \) such that the points \( A(5, 5, \lambda) \), \( B(-1, 3, 2) \), and \( C(-4, 2, -2) \) are collinear, we can use the concept of direction ratios.

### Step-by-Step Solution:

1. **Find the Direction Ratios of Vectors AB and BC**:
- The direction ratios of vector \( AB \) can be calculated as:
\[ AB = B - A = (-1 - 5, 3 - 5, 2 - \lambda) = (-6, -2, 2 - \lambda) \]
- The direction ratios of vector \( BC \) can be calculated as:
\[ BC = C - B = (-4 - (-1), 2 - 3, -2 - 2) = (-3, -1, -4) \]

2. **Set Up the Proportionality Condition**:
- Since points A, B, and C are collinear, the direction ratios of \( AB \) and \( BC \) must be proportional. This gives us the following equations:
\[ \frac{-6}{-3} = \frac{-2}{-1} = \frac{2 - \lambda}{-4} \]

3. **Simplify the Ratios**:
- Simplifying the first two ratios:
\[ \frac{-6}{-3} = 2 \quad \text{and} \quad \frac{-2}{-1} = 2 \]
- Therefore, we can equate the third ratio:
\[ 2 = \frac{2 - \lambda}{-4} \]

4. **Cross-Multiply to Solve for \( \lambda \)**:
- Cross-multiplying gives:
\[ 2 \cdot (-4) = 2 - \lambda \]
\[ -8 = 2 - \lambda \]

5. **Rearranging the Equation**:
- Rearranging the equation to solve for \( \lambda \):
\[ -\lambda = -8 - 2 \]
\[ -\lambda = -10 \]
\[ \lambda = 10 \]

### Final Answer:
The value of \( \lambda \) is \( 10 \).