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

Options

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

Explanation

To determine whether the points represented by the position vectors \(\vec{P} = \vec{a} - 2\vec{b} + 3\vec{c}\), \(\vec{Q} = 2\vec{a} + 3\vec{b} - 4\vec{c}\), and \(\vec{R} = -7\vec{b} + 10\vec{c}\) are collinear, we can follow these steps:

### Step 1: Define the position vectors
Let:
- \(\vec{P} = \vec{a} - 2\vec{b} + 3\vec{c}\)
- \(\vec{Q} = 2\vec{a} + 3\vec{b} - 4\vec{c}\)
- \(\vec{R} = -7\vec{b} + 10\vec{c}\)

### Step 2: Find the vector \(\vec{QR}\)
The vector \(\vec{QR}\) is given by:
\[ \vec{QR} = \vec{R} - \vec{Q} \]
Substituting the values:
\[ \vec{QR} = (-7\vec{b} + 10\vec{c}) - (2\vec{a} + 3\vec{b} - 4\vec{c}) \]
\[ = -7\vec{b} + 10\vec{c} - 2\vec{a} - 3\vec{b} + 4\vec{c} \]
\[ = -2\vec{a} - 10\vec{b} + 14\vec{c} \]

### Step 3: Find the vector \(\vec{PQ}\)
The vector \(\vec{PQ}\) is given by:
\[ \vec{PQ} = \vec{Q} - \vec{P} \]
Substituting the values:
\[ \vec{PQ} = (2\vec{a} + 3\vec{b} - 4\vec{c}) - (\vec{a} - 2\vec{b} + 3\vec{c}) \]
\[ = 2\vec{a} + 3\vec{b} - 4\vec{c} - \vec{a} + 2\vec{b} - 3\vec{c} \]
\[ = \vec{a} + 5\vec{b} - 7\vec{c} \]

### Step 4: Check for collinearity
To check if points \(P\), \(Q\), and \(R\) are collinear, we need to see if \(\vec{QR}\) is a scalar multiple of \(\vec{PQ}\):
\[ \vec{QR} = -2\vec{a} - 10\vec{b} + 14\vec{c} \]
\[ \vec{PQ} = \vec{a} + 5\vec{b} - 7\vec{c} \]
We can express \(\vec{QR}\) as:
\[ \vec{QR} = -2(\vec{PQ}) \]
This shows that \(\vec{QR}\) is indeed a scalar multiple of \(\vec{PQ}\).

### Conclusion
Since \(\vec{QR} = -2 \vec{PQ}\), the points \(P\), \(Q\), and \(R\) are collinear.