To find the coordinates of the points that divide the line segment AB into four equal parts, we will follow these steps:
### Step 1: Identify the Coordinates of Points A and B
Let the coordinates of point A be \( A(-2, 0, 6) \) and the coordinates of point B be \( B(10, -6, -12) \).
### Step 2: Determine the Number of Equal Parts
Since we need to divide the line segment AB into four equal parts, we will find three points that divide the segment into four segments.
### Step 3: Calculate the Coordinates of the Division Points
The division points can be found using the section formula. The section formula states that if a point divides the line segment joining two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in the ratio \( m:n \), then the coordinates of the point are given by:
\[
\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right)
\]
In our case, since we are dividing the segment into four equal parts, the ratios will be \( 1:3 \), \( 2:2 \) (or \( 1:1 \)), and \( 3:1 \).
### Step 4: Calculate the First Division Point (1:3)
Using the section formula for the first point \( P_1 \) which divides \( AB \) in the ratio \( 1:3 \):
\[
P_1 = \left( \frac{1 \cdot 10 + 3 \cdot (-2)}{1 + 3}, \frac{1 \cdot (-6) + 3 \cdot 0}{1 + 3}, \frac{1 \cdot (-12) + 3 \cdot 6}{1 + 3} \right)
\]
Calculating each coordinate:
- \( x \) coordinate:
\[
\frac{10 - 6}{4} = \frac{4}{4} = 1
\]
- \( y \) coordinate:
\[
\frac{-6 + 0}{4} = \frac{-6}{4} = -\frac{3}{2}
\]
- \( z \) coordinate:
\[
\frac{-12 + 18}{4} = \frac{6}{4} = \frac{3}{2}
\]
Thus, the first division point \( P_1 \) is \( (1, -\frac{3}{2}, \frac{3}{2}) \).
### Step 5: Calculate the Second Division Point (2:2)
For the second point \( P_2 \) which divides \( AB \) in the ratio \( 1:1 \):
\[
P_2 = \left( \frac{1 \cdot 10 + 1 \cdot (-2)}{1 + 1}, \frac{1 \cdot (-6) + 1 \cdot 0}{1 + 1}, \frac{1 \cdot (-12) + 1 \cdot 6}{1 + 1} \right)
\]
Calculating each coordinate:
- \( x \) coordinate:
\[
\frac{10 - 2}{2} = \frac{8}{2} = 4
\]
- \( y \) coordinate:
\[
\frac{-6 + 0}{2} = \frac{-6}{2} = -3
\]
- \( z \) coordinate:
\[
\frac{-12 + 6}{2} = \frac{-6}{2} = -3
\]
Thus, the second division point \( P_2 \) is \( (4, -3, -3) \).
### Step 6: Calculate the Third Division Point (3:1)
For the third point \( P_3 \) which divides \( AB \) in the ratio \( 3:1 \):
\[
P_3 = \left( \frac{3 \cdot 10 + 1 \cdot (-2)}{3 + 1}, \frac{3 \cdot (-6) + 1 \cdot 0}{3 + 1}, \frac{3 \cdot (-12) + 1 \cdot 6}{3 + 1} \right)
\]
Calculating each coordinate:
- \( x \) coordinate:
\[
\frac{30 - 2}{4} = \frac{28}{4} = 7
\]
- \( y \) coordinate:
\[
\frac{-18 + 0}{4} = \frac{-18}{4} = -\frac{9}{2}
\]
- \( z \) coordinate:
\[
\frac{-36 + 6}{4} = \frac{-30}{4} = -\frac{15}{2}
\]
Thus, the third division point \( P_3 \) is \( (7, -\frac{9}{2}, -\frac{15}{2}) \).
### Final Result
The coordinates of the points that divide the line segment AB into four equal parts are:
1. \( P_1(1, -\frac{3}{2}, \frac{3}{2}) \)
2. \( P_2(4, -3, -3) \)
3. \( P_3(7, -\frac{9}{2}, -\frac{15}{2}) \)
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