If the points (a, b, c), (2, 0, -1) and (1, -1, -3) are
collinear, then the value of a is

Options

A. 2
B. 4
C. 3
D. 1

Answer

To determine the value of \( a \) such that the points \( (a, b, c) \), \( (2, 0, -1) \), and \( (1, -1, -3) \) are collinear, we can follow these steps:

### Step 1: Identify the Points
Let:
- Point A: \( (a, b, c) \)
- Point B: \( (2, 0, -1) \)
- Point C: \( (1, -1, -3) \)

### Step 2: Find Direction Ratios
For the points to be collinear, the direction ratios of vector \( AB \) must be proportional to the direction ratios of vector \( BC \).

**Direction Ratios of \( AB \)**:
\[ AB = B - A = (2 - a, 0 - b, -1 - c) \]

**Direction Ratios of \( BC \)**:
\[ BC = C - B = (1 - 2, -1 - 0, -3 - (-1)) = (-1, -1, -2) \]

### Step 3: Set Up the Proportionality Condition
Since the points are collinear, we can set the direction ratios equal:
\[ (2 - a, 0 - b, -1 - c) \propto (-1, -1, -2) \]

This gives us the following equations:
1. \( 2 - a = k(-1) \) for some scalar \( k \)
2. \( 0 - b = k(-1) \)
3. \( -1 - c = k(-2) \)

### Step 4: Solve for \( a \)
From the first equation:
\[ 2 - a = -k \implies a = 2 + k \]

### Step 5: Solve for \( b \)
From the second equation:
\[ -b = -k \implies b = k \]

### Step 6: Solve for \( c \)
From the third equation:
\[ -1 - c = -2k \implies c = -1 + 2k \]

### Step 7: Express \( a \) in terms of \( k \)
Now we have:
- \( a = 2 + k \)
- \( b = k \)
- \( c = -1 + 2k \)

### Step 8: Find a Specific Value for \( k \)
To find a specific value for \( a \), we can choose \( k = 1 \) (as an example):
- \( a = 2 + 1 = 3 \)
- \( b = 1 \)
- \( c = -1 + 2(1) = 1 \)

Thus, the value of \( a \) is:
\[ \boxed{3} \]