Are the points (-1,4,-2),(2,-2,1) and (0,2,-1) collinear

Answer

To determine whether the points A(-1, 4, -2), B(2, -2, 1), and C(0, 2, -1) are collinear, we can use the concept of the section formula. Here are the steps to solve the problem:

### Step 1: Identify the Points
Let:
- A = (-1, 4, -2)
- B = (2, -2, 1)
- C = (0, 2, -1)

### Step 2: Use the Section Formula
Assume point B divides the line segment AC in the ratio λ:1. According to the section formula, the coordinates of point B can be expressed as:
- \( x_B = \frac{\lambda \cdot x_C + 1 \cdot x_A}{\lambda + 1} \)
- \( y_B = \frac{\lambda \cdot y_C + 1 \cdot y_A}{\lambda + 1} \)
- \( z_B = \frac{\lambda \cdot z_C + 1 \cdot z_A}{\lambda + 1} \)

### Step 3: Substitute the Coordinates
Substituting the coordinates of points A and C into the formulas:
- For x-coordinate:
\[ 2 = \frac{\lambda \cdot 0 + 1 \cdot (-1)}{\lambda + 1} \implies 2(\lambda + 1) = -1 \implies 2\lambda + 2 = -1 \implies 2\lambda = -3 \implies \lambda = -\frac{3}{2} \]

- For y-coordinate:
\[ -2 = \frac{\lambda \cdot 2 + 1 \cdot 4}{\lambda + 1} \implies -2(\lambda + 1) = 2\lambda + 4 \implies -2\lambda - 2 = 2\lambda + 4 \implies -4\lambda = 6 \implies \lambda = -\frac{3}{2} \]

- For z-coordinate:
\[ 1 = \frac{\lambda \cdot (-1) + 1 \cdot (-2)}{\lambda + 1} \implies 1(\lambda + 1) = -\lambda - 2 \implies \lambda + 1 = -\lambda - 2 \implies 2\lambda = -3 \implies \lambda = -\frac{3}{2} \]

### Step 4: Conclusion
Since we found the same value of λ = -3/2 from all three coordinates, it indicates that points A, B, and C are collinear.

### Final Answer
The points (-1, 4, -2), (2, -2, 1), and (0, 2, -1) are collinear.

---