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

Answer

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 B divides the line segment joining points A (x1, y1, z1) and C (x2, y2, z2) in the ratio m:n, then the coordinates of point B can be given by:

\[ B\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) \]

### Step 2: Assign Coordinates
Let:
- A = (-2, 3, 5) → (x1, y1, z1)
- B = (1, 2, 3) → (x, y, z)
- C = (7, 0, -1) → (x2, y2, z2)

### Step 3: Assume a Ratio
Assume that point B divides the line segment AC in the ratio λ:1. Thus, we can express the coordinates of B in terms of λ:

\[ B\left(\frac{7\lambda - 2}{\lambda + 1}, \frac{0\lambda + 3}{\lambda + 1}, \frac{-\lambda + 5}{\lambda + 1}\right) \]

### Step 4: Set Up the Equations
Now, we can set up equations for each coordinate based on the coordinates of point B:

1. For the x-coordinate:
\[ \frac{7\lambda - 2}{\lambda + 1} = 1 \]

2. For the y-coordinate:
\[ \frac{3}{\lambda + 1} = 2 \]

3. For the z-coordinate:
\[ \frac{-\lambda + 5}{\lambda + 1} = 3 \]

### Step 5: Solve for λ
**From the x-coordinate:**
\[ 7\lambda - 2 = \lambda + 1 \\ 7\lambda - \lambda = 1 + 2 \\ 6\lambda = 3 \\ \lambda = \frac{1}{2} \]

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

**From the z-coordinate:**
\[ -\lambda + 5 = 3(\lambda + 1) \\ -\lambda + 5 = 3\lambda + 3 \\ 5 - 3 = 3\lambda + \lambda \\ 2 = 4\lambda \\ \lambda = \frac{1}{2} \]

### Step 6: Conclusion
Since we found the same value of λ = 1/2 from all three coordinates, it confirms that point B divides the line segment AC in the ratio 1:2. Therefore, points A, B, and C are collinear.