If the position vector of three points are a - 2b + 3c, 2a + 3b - 4c, - 7b + 10 c, then the three points are

Options

A. collinear
B. non-coplanar
C. non-collinear
D. None of these

Explanation

To determine the relationship between the three points represented by the position vectors \( \mathbf{P} = \mathbf{a} - 2\mathbf{b} + 3\mathbf{c} \), \( \mathbf{Q} = 2\mathbf{a} + 3\mathbf{b} - 4\mathbf{c} \), and \( \mathbf{R} = -7\mathbf{b} + 10\mathbf{c} \), we will analyze the vectors formed by these points.

### Step 1: Find the vectors \( \mathbf{PQ} \) and \( \mathbf{QR} \)

1. **Calculate \( \mathbf{PQ} \)**:
\[ \mathbf{PQ} = \mathbf{Q} - \mathbf{P} = (2\mathbf{a} + 3\mathbf{b} - 4\mathbf{c}) - (\mathbf{a} - 2\mathbf{b} + 3\mathbf{c}) \]
Simplifying this:
\[ \mathbf{PQ} = (2\mathbf{a} - \mathbf{a}) + (3\mathbf{b} + 2\mathbf{b}) + (-4\mathbf{c} - 3\mathbf{c}) = \mathbf{a} + 5\mathbf{b} - 7\mathbf{c} \]

2. **Calculate \( \mathbf{QR} \)**:
\[ \mathbf{QR} = \mathbf{R} - \mathbf{Q} = (-7\mathbf{b} + 10\mathbf{c}) - (2\mathbf{a} + 3\mathbf{b} - 4\mathbf{c}) \]
Simplifying this:
\[ \mathbf{QR} = (-7\mathbf{b} - 3\mathbf{b}) + (10\mathbf{c} + 4\mathbf{c}) - 2\mathbf{a} = -2\mathbf{a} - 10\mathbf{b} + 14\mathbf{c} \]

### Step 2: Check for collinearity

To check if the points are collinear, we need to see if \( \mathbf{QR} \) is a scalar multiple of \( \mathbf{PQ} \). We can express this as:
\[ \mathbf{QR} = k \cdot \mathbf{PQ} \]
for some scalar \( k \).

From our calculations:
- \( \mathbf{PQ} = \mathbf{a} + 5\mathbf{b} - 7\mathbf{c} \)
- \( \mathbf{QR} = -2\mathbf{a} - 10\mathbf{b} + 14\mathbf{c} \)

### Step 3: Set up the equations

Assuming \( \mathbf{QR} = k \cdot \mathbf{PQ} \):
\[ -2\mathbf{a} - 10\mathbf{b} + 14\mathbf{c} = k(\mathbf{a} + 5\mathbf{b} - 7\mathbf{c}) \]
This gives us the following system of equations:
1. \( -2 = k \)
2. \( -10 = 5k \)
3. \( 14 = -7k \)

### Step 4: Solve for \( k \)

From the first equation, we have:
\[ k = -2 \]

Substituting \( k = -2 \) into the second equation:
\[ -10 = 5(-2) \implies -10 = -10 \quad \text{(True)} \]

Substituting \( k = -2 \) into the third equation:
\[ 14 = -7(-2) \implies 14 = 14 \quad \text{(True)} \]

Since all equations are satisfied, \( \mathbf{QR} \) is indeed a scalar multiple of \( \mathbf{PQ} \).

### Conclusion

The points \( P, Q, R \) are collinear.