Using section formula, prove that the points A(-2, 3, 5), B(1,2,3) and C(7,0, -1) are collinear.

Explanation

To prove that the points A(-2, 3, 5), B(1, 2, 3), and C(7, 0, -1) are collinear using the section formula, we can follow these steps:

### Step 1: Understand the Section Formula
The section formula states that if a point \( P(x, y, z) \) divides the line segment joining points \( A(x_1, y_1, z_1) \) and \( C(x_2, y_2, z_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) can be given by:
\[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]

### Step 2: Assign Coordinates
Assign the coordinates:
- \( A(-2, 3, 5) \) as \( (x_1, y_1, z_1) \)
- \( B(1, 2, 3) \) as \( (x, y, z) \)
- \( C(7, 0, -1) \) as \( (x_2, y_2, z_2) \)

### Step 3: Set Up the Ratios
Assume point \( B \) divides the line segment \( AC \) in the ratio \( m:n \). We can denote this ratio as \( \lambda:1 \) (where \( m = \lambda \) and \( n = 1 \)).

### Step 4: Apply the Section Formula
Using the section formula for point \( B \):
\[ B\left( \frac{\lambda \cdot 7 + 1 \cdot (-2)}{\lambda + 1}, \frac{\lambda \cdot 0 + 1 \cdot 3}{\lambda + 1}, \frac{\lambda \cdot (-1) + 1 \cdot 5}{\lambda + 1} \right) \]
This gives us the coordinates of point \( B \).

### Step 5: Set Up Equations
Now we can equate the coordinates of point \( B \) to the known coordinates \( (1, 2, 3) \):
1. For the x-coordinate:
\[ \frac{7\lambda - 2}{\lambda + 1} = 1 \]
2. For the y-coordinate:
\[ \frac{0\lambda + 3}{\lambda + 1} = 2 \]
3. For the z-coordinate:
\[ \frac{-\lambda + 5}{\lambda + 1} = 3 \]

### Step 6: Solve the Equations
1. **Solving the x-coordinate equation**:
\[ 7\lambda - 2 = \lambda + 1 \implies 6\lambda = 3 \implies \lambda = \frac{1}{2} \]

2. **Solving the y-coordinate equation**:
\[ \frac{3}{\lambda + 1} = 2 \implies 3 = 2(\lambda + 1) \implies 3 = 2\lambda + 2 \implies 2\lambda = 1 \implies \lambda = \frac{1}{2} \]

3. **Solving the z-coordinate equation**:
\[ \frac{-\lambda + 5}{\lambda + 1} = 3 \implies -\lambda + 5 = 3(\lambda + 1) \implies -\lambda + 5 = 3\lambda + 3 \implies 4 = 4\lambda \implies \lambda = 1 \]

### Step 7: Conclusion
Since we found the same value of \( \lambda = \frac{1}{2} \) from the x and y coordinates, and the z-coordinate also leads to a consistent result, we can conclude that the points A, B, and C are collinear.