If the points `A(1, -2, 3), B(2, 3, -4) and C(0, -p, 10)` are collinear, then p=

Options

A. `7`
B. `-7`
C. `5`
D. `-5`

Answer

To determine the value of \( p \) such that the points \( A(1, -2, 3) \), \( B(2, 3, -4) \), and \( C(0, -p, 10) \) are collinear, we will follow these steps:

### Step 1: Find the vectors \( \vec{AB} \) and \( \vec{BC} \)

1. **Calculate \( \vec{AB} \)**:
\[ \vec{AB} = \vec{B} - \vec{A} = (2 - 1, 3 - (-2), -4 - 3) = (1, 5, -7) \]

2. **Calculate \( \vec{BC} \)**:
\[ \vec{BC} = \vec{C} - \vec{B} = (0 - 2, -p - 3, 10 - (-4)) = (-2, -p - 3, 14) \]

### Step 2: Set up the condition for collinearity

For points \( A \), \( B \), and \( C \) to be collinear, the vectors \( \vec{AB} \) and \( \vec{BC} \) must be parallel. This means that the ratios of their corresponding components must be equal:
\[ \frac{1}{-2} = \frac{5}{-p - 3} = \frac{-7}{14} \]

### Step 3: Simplify the ratios

1. **From the third ratio**:
\[ \frac{-7}{14} = -\frac{1}{2} \]
This implies:
\[ \frac{1}{-2} = -\frac{1}{2} \quad \text{(this is true)} \]

2. **Now, set the second ratio equal to \(-\frac{1}{2}\)**:
\[ \frac{5}{-p - 3} = -\frac{1}{2} \]

### Step 4: Cross-multiply to solve for \( p \)

Cross-multiplying gives:
\[ 5 \cdot 2 = -1 \cdot (-p - 3) \]
\[ 10 = p + 3 \]

### Step 5: Solve for \( p \)

Rearranging the equation:
\[ p = 10 - 3 = 7 \]

### Conclusion

Thus, the value of \( p \) is \( 7 \).