The points `a-2b+3c, 2a+3b-4c, -7b+10 c` are

Options

A. collinear
B. non-collinear
C. can't say
D. None of these

Explanation

To determine whether the points \( A = a - 2b + 3c \), \( B = 2a + 3b - 4c \), and \( C = -7b + 10c \) are collinear, we can follow these steps:

### Step 1: Define the Points
We have three points defined as:
- \( A = a - 2b + 3c \)
- \( B = 2a + 3b - 4c \)
- \( C = -7b + 10c \)

### Step 2: Find Vectors AB and AC
To check for collinearity, we will find the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \).

**Vector \( \overrightarrow{AB} \):**
\[ \overrightarrow{AB} = B - A = (2a + 3b - 4c) - (a - 2b + 3c) \]
Calculating this gives:
\[ = 2a + 3b - 4c - a + 2b - 3c \]
\[ = (2a - a) + (3b + 2b) + (-4c - 3c) \]
\[ = a + 5b - 7c \]

**Vector \( \overrightarrow{AC} \):**
\[ \overrightarrow{AC} = C - A = (-7b + 10c) - (a - 2b + 3c) \]
Calculating this gives:
\[ = -7b + 10c - a + 2b - 3c \]
\[ = -a + (-7b + 2b) + (10c - 3c) \]
\[ = -a - 5b + 7c \]

### Step 3: Check for Collinearity
The points \( A \), \( B \), and \( C \) are collinear if the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are scalar multiples of each other. This means we can express \( \overrightarrow{AB} \) as a scalar multiple of \( \overrightarrow{AC} \).

We have:
\[ \overrightarrow{AB} = a + 5b - 7c \]
\[ \overrightarrow{AC} = -a - 5b + 7c \]

To check if \( \overrightarrow{AB} = k \cdot \overrightarrow{AC} \) for some scalar \( k \):
\[ a + 5b - 7c = k(-a - 5b + 7c) \]

### Step 4: Set Up the Equations
Equating the coefficients gives us:
1. \( 1 = -k \) (for \( a \))
2. \( 5 = -5k \) (for \( b \))
3. \( -7 = 7k \) (for \( c \))

From the first equation, \( k = -1 \).
From the second equation, substituting \( k = -1 \) gives \( 5 = 5 \), which is true.
From the third equation, substituting \( k = -1 \) gives \( -7 = -7 \), which is also true.

### Conclusion
Since we found a consistent value for \( k \) that satisfies all equations, the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are indeed scalar multiples of each other. Therefore, the points \( A \), \( B \), and \( C \) are collinear.

**Final Answer:** The points are collinear.

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