The three points A (0,0,0), B (2,-3, 3), C(-2,3,-3) are collinear. Find in what ratio each point divides the segment joining the other two.

Answer

To find the ratio in which each point divides the segment joining the other two points, we will follow these steps:

### Step 1: Identify the Points
The given points are:
- A(0, 0, 0)
- B(2, -3, 3)
- C(-2, 3, -3)

### Step 2: Calculate the Distances
We will calculate the distances between the points A, B, and C using the distance formula in 3D space, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

#### Distance AB
Using points A(0, 0, 0) and B(2, -3, 3):
\[ AB = \sqrt{(2 - 0)^2 + (-3 - 0)^2 + (3 - 0)^2} = \sqrt{2^2 + (-3)^2 + 3^2} = \sqrt{4 + 9 + 9} = \sqrt{22} \]

#### Distance BC
Using points B(2, -3, 3) and C(-2, 3, -3):
\[ BC = \sqrt{(-2 - 2)^2 + (3 - (-3))^2 + (-3 - 3)^2} = \sqrt{(-4)^2 + (6)^2 + (-6)^2} = \sqrt{16 + 36 + 36} = \sqrt{88} \]

#### Distance AC
Using points A(0, 0, 0) and C(-2, 3, -3):
\[ AC = \sqrt{(-2 - 0)^2 + (3 - 0)^2 + (-3 - 0)^2} = \sqrt{(-2)^2 + (3)^2 + (-3)^2} = \sqrt{4 + 9 + 9} = \sqrt{22} \]

### Step 3: Determine the Ratios
Now, we will find the ratio in which point A divides segment BC. We will use the section formula. Let the ratio be \( m:n \).

Using the section formula in 3D:
\[ x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n}, \quad z = \frac{mz_2 + nz_1}{m+n} \]

For point A(0, 0, 0) dividing segment BC:
- \( x_1 = -2, y_1 = 3, z_1 = -3 \) (coordinates of C)
- \( x_2 = 2, y_2 = -3, z_2 = 3 \) (coordinates of B)

Setting up the equations for x, y, and z coordinates:
1. For x-coordinate:
\[ 0 = \frac{m(2) + n(-2)}{m+n} \implies 0 = 2m - 2n \implies m = n \]

2. For y-coordinate:
\[ 0 = \frac{m(-3) + n(3)}{m+n} \implies 0 = -3m + 3n \implies m = n \]

3. For z-coordinate:
\[ 0 = \frac{m(3) + n(-3)}{m+n} \implies 0 = 3m - 3n \implies m = n \]

### Conclusion
From all three coordinates, we find that \( m = n \). Therefore, the ratio in which point A divides segment BC is:
\[ \text{Ratio} = 1:1 \]