If the points `(-1, -1, 2), (2,m,5) and (3,11,6)` are collinear, find the value of m.

Answer

To find the value of \( m \) such that the points \((-1, -1, 2)\), \((2, m, 5)\), and \((3, 11, 6)\) are collinear, we can follow these steps:

### Step 1: Define the Points
Let:
- Point A = \((-1, -1, 2)\)
- Point B = \((2, m, 5)\)
- Point C = \((3, 11, 6)\)

### Step 2: Find Direction Ratios
To determine if the points are collinear, we need to find the direction ratios of vectors \( \overrightarrow{AB} \) and \( \overrightarrow{BC} \).

#### Calculate \( \overrightarrow{AB} \):
\[ \overrightarrow{AB} = B - A = (2 - (-1), m - (-1), 5 - 2) = (2 + 1, m + 1, 5 - 2) = (3, m + 1, 3) \]

#### Calculate \( \overrightarrow{BC} \):
\[ \overrightarrow{BC} = C - B = (3 - 2, 11 - m, 6 - 5) = (1, 11 - m, 1) \]

### Step 3: Set Up Proportionality Condition
For the points to be collinear, the direction ratios of \( \overrightarrow{AB} \) and \( \overrightarrow{BC} \) must be proportional. This means:
\[ \frac{3}{1} = \frac{m + 1}{11 - m} = \frac{3}{1} \]

### Step 4: Solve the Proportionality Condition
From the first part of the proportion:
\[ \frac{3}{1} = \frac{m + 1}{11 - m} \]
Cross-multiplying gives:
\[ 3(11 - m) = 1(m + 1) \]
Expanding both sides:
\[ 33 - 3m = m + 1 \]
Rearranging the equation:
\[ 33 - 1 = m + 3m \]
\[ 32 = 4m \]
Dividing both sides by 4:
\[ m = \frac{32}{4} = 8 \]

### Conclusion
The value of \( m \) is \( 8 \).

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