Find the coordinates of the point which trisect the line segment joining `A(2,1,-3)` and `B(5,-8,3)`.

Answer

To find the coordinates of the points that trisect the line segment joining points A(2, 1, -3) and B(5, -8, 3), we will follow these steps:

### Step 1: Understand the Trisection
The points that trisect the line segment AB are P and Q. Point P divides the segment in the ratio 1:2, and point Q divides it in the ratio 2:1.

### Step 2: Use the Section Formula
The section formula for a point dividing a line segment in the ratio m:n is given by:

\[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]

Where:
- \( (x_1, y_1, z_1) \) are the coordinates of point A
- \( (x_2, y_2, z_2) \) are the coordinates of point B

### Step 3: Calculate Coordinates of Point P
For point P, we have:
- \( m = 1 \)
- \( n = 2 \)
- \( A(2, 1, -3) \) and \( B(5, -8, 3) \)

Now, substituting into the section formula:

\[ P = \left( \frac{1 \cdot 5 + 2 \cdot 2}{1+2}, \frac{1 \cdot (-8) + 2 \cdot 1}{1+2}, \frac{1 \cdot 3 + 2 \cdot (-3)}{1+2} \right) \]

Calculating each coordinate:
- For x-coordinate:
\[ x_P = \frac{5 + 4}{3} = \frac{9}{3} = 3 \]

- For y-coordinate:
\[ y_P = \frac{-8 + 2}{3} = \frac{-6}{3} = -2 \]

- For z-coordinate:
\[ z_P = \frac{3 - 6}{3} = \frac{-3}{3} = -1 \]

Thus, the coordinates of point P are \( P(3, -2, -1) \).

### Step 4: Calculate Coordinates of Point Q
For point Q, we have:
- \( m = 2 \)
- \( n = 1 \)

Now substituting into the section formula:

\[ Q = \left( \frac{2 \cdot 5 + 1 \cdot 2}{2+1}, \frac{2 \cdot (-8) + 1 \cdot 1}{2+1}, \frac{2 \cdot 3 + 1 \cdot (-3)}{2+1} \right) \]

Calculating each coordinate:
- For x-coordinate:
\[ x_Q = \frac{10 + 2}{3} = \frac{12}{3} = 4 \]

- For y-coordinate:
\[ y_Q = \frac{-16 + 1}{3} = \frac{-15}{3} = -5 \]

- For z-coordinate:
\[ z_Q = \frac{6 - 3}{3} = \frac{3}{3} = 1 \]

Thus, the coordinates of point Q are \( Q(4, -5, 1) \).

### Final Answer
The coordinates of the points that trisect the line segment joining A(2, 1, -3) and B(5, -8, 3) are:
- Point P: \( (3, -2, -1) \)
- Point Q: \( (4, -5, 1) \)