To find the coordinates of the points that trisect the line segment joining the points \( P(4, 2, -6) \) and \( Q(10, -16, 6) \), we will use the section formula. The points that trisect the line segment divide it into three equal parts. Let's denote the points that trisect the segment as \( R \) and \( S \).
### Step 1: Identify the points and the ratios
The points \( P \) and \( Q \) are given as:
- \( P(4, 2, -6) \)
- \( Q(10, -16, 6) \)
Since we want to trisect the line segment, we will divide it into three equal parts. The point \( R \) divides the segment \( PQ \) in the ratio \( 1:2 \) and the point \( S \) divides the segment \( PQ \) in the ratio \( 2:1 \).
### Step 2: Use the section formula for point \( R \)
The section formula for a point dividing a line segment in the ratio \( m:n \) is given by:
\[
\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right)
\]
For point \( R \) (ratio \( 1:2 \)):
- \( m = 1 \), \( n = 2 \)
- \( (x_1, y_1, z_1) = (4, 2, -6) \)
- \( (x_2, y_2, z_2) = (10, -16, 6) \)
Calculating the coordinates of \( R \):
- \( x_R = \frac{1 \cdot 10 + 2 \cdot 4}{1 + 2} = \frac{10 + 8}{3} = \frac{18}{3} = 6 \)
- \( y_R = \frac{1 \cdot (-16) + 2 \cdot 2}{1 + 2} = \frac{-16 + 4}{3} = \frac{-12}{3} = -4 \)
- \( z_R = \frac{1 \cdot 6 + 2 \cdot (-6)}{1 + 2} = \frac{6 - 12}{3} = \frac{-6}{3} = -2 \)
Thus, the coordinates of point \( R \) are \( (6, -4, -2) \).
### Step 3: Use the section formula for point \( S \)
For point \( S \) (ratio \( 2:1 \)):
- \( m = 2 \), \( n = 1 \)
Calculating the coordinates of \( S \):
- \( x_S = \frac{2 \cdot 10 + 1 \cdot 4}{2 + 1} = \frac{20 + 4}{3} = \frac{24}{3} = 8 \)
- \( y_S = \frac{2 \cdot (-16) + 1 \cdot 2}{2 + 1} = \frac{-32 + 2}{3} = \frac{-30}{3} = -10 \)
- \( z_S = \frac{2 \cdot 6 + 1 \cdot (-6)}{2 + 1} = \frac{12 - 6}{3} = \frac{6}{3} = 2 \)
Thus, the coordinates of point \( S \) are \( (8, -10, 2) \).
### Final Answer
The coordinates of the points that trisect the line segment joining \( P \) and \( Q \) are:
- Point \( R(6, -4, -2) \)
- Point \( S(8, -10, 2) \)
