Find the co-ordinates of the points of trisection of the line segment joining the points A(2, -3, 5) and B(6, 0, -1).

Explanation

To find the coordinates of the points of trisection of the line segment joining the points A(2, -3, 5) and B(6, 0, -1), we will follow these steps:

### Step 1: Understand the concept of trisection
Trisection means dividing the line segment into three equal parts. Therefore, we need to find two points (let's call them C and D) that divide the segment AB into three equal segments.

### Step 2: Use the section formula
The section formula in three-dimensional geometry states that if a point divides the line segment joining points \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\) in the ratio \(m:n\), then the coordinates of the point are given by:
\[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]

### Step 3: Find the coordinates of point C (1:2 ratio)
For point C, which divides the segment in the ratio \(1:2\):
- \(A(2, -3, 5)\) and \(B(6, 0, -1)\)
- Here, \(m = 1\) and \(n = 2\).

Using the section formula:
\[ C_x = \frac{1 \cdot 6 + 2 \cdot 2}{1 + 2} = \frac{6 + 4}{3} = \frac{10}{3} \]
\[ C_y = \frac{1 \cdot 0 + 2 \cdot (-3)}{1 + 2} = \frac{0 - 6}{3} = -2 \]
\[ C_z = \frac{1 \cdot (-1) + 2 \cdot 5}{1 + 2} = \frac{-1 + 10}{3} = \frac{9}{3} = 3 \]
Thus, the coordinates of point C are \(\left(\frac{10}{3}, -2, 3\right)\).

### Step 4: Find the coordinates of point D (2:1 ratio)
For point D, which divides the segment in the ratio \(2:1\):
- Here, \(m = 2\) and \(n = 1\).

Using the section formula:
\[ D_x = \frac{2 \cdot 6 + 1 \cdot 2}{2 + 1} = \frac{12 + 2}{3} = \frac{14}{3} \]
\[ D_y = \frac{2 \cdot 0 + 1 \cdot (-3)}{2 + 1} = \frac{0 - 3}{3} = -1 \]
\[ D_z = \frac{2 \cdot (-1) + 1 \cdot 5}{2 + 1} = \frac{-2 + 5}{3} = \frac{3}{3} = 1 \]
Thus, the coordinates of point D are \(\left(\frac{14}{3}, -1, 1\right)\).

### Final Result
The coordinates of the points of trisection of the line segment joining points A and B are:
- Point C: \(\left(\frac{10}{3}, -2, 3\right)\)
- Point D: \(\left(\frac{14}{3}, -1, 1\right)\)