If the points with position vectors ` 20 hati + p hatj , 5 hati - hatj and 10 hati - 13 hatj` are collinear, then p =

Options

A. 7
B. -37
C. -7
D. 37

Explanation

To determine the value of \( p \) such that the points with position vectors \( \mathbf{A} = 20 \hat{i} + p \hat{j} \), \( \mathbf{B} = 5 \hat{i} - \hat{j} \), and \( \mathbf{C} = 10 \hat{i} - 13 \hat{j} \) are collinear, we can follow these steps:

### Step 1: Define the vectors
Let:
- \( \mathbf{A} = 20 \hat{i} + p \hat{j} \)
- \( \mathbf{B} = 5 \hat{i} - 1 \hat{j} \)
- \( \mathbf{C} = 10 \hat{i} - 13 \hat{j} \)

### Step 2: Find the vectors \( \mathbf{AB} \) and \( \mathbf{BC} \)
The vector \( \mathbf{AB} \) can be calculated as:
\[ \mathbf{AB} = \mathbf{B} - \mathbf{A} = (5 \hat{i} - 1 \hat{j}) - (20 \hat{i} + p \hat{j}) = (5 - 20) \hat{i} + (-1 - p) \hat{j} = -15 \hat{i} + (-1 - p) \hat{j} \]

The vector \( \mathbf{BC} \) can be calculated as:
\[ \mathbf{BC} = \mathbf{C} - \mathbf{B} = (10 \hat{i} - 13 \hat{j}) - (5 \hat{i} - 1 \hat{j}) = (10 - 5) \hat{i} + (-13 + 1) \hat{j} = 5 \hat{i} - 12 \hat{j} \]

### Step 3: Set up the collinearity condition
For the points to be collinear, the vectors \( \mathbf{AB} \) and \( \mathbf{BC} \) must be scalar multiples of each other. This means:
\[ \mathbf{AB} = k \mathbf{BC} \]
for some scalar \( k \).

### Step 4: Compare coefficients
From the expressions for \( \mathbf{AB} \) and \( \mathbf{BC} \), we can write:
\[ -15 \hat{i} + (-1 - p) \hat{j} = k (5 \hat{i} - 12 \hat{j}) \]

This gives us two equations:
1. For \( \hat{i} \):
\[ -15 = 5k \implies k = -3 \]
2. For \( \hat{j} \):
\[ -1 - p = -12k \]

Substituting \( k = -3 \) into the second equation:
\[ -1 - p = -12(-3) = 36 \]
\[ -1 - p = 36 \]
\[ -p = 36 + 1 = 37 \]
\[ p = -37 \]

### Final Answer
Thus, the value of \( p \) is:
\[ \boxed{-37} \]