Find the coordinates of the points which trisect AB given that A (2, 1, -3) and B (5,.- 8,3).

Answer

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

### Step 1: Understand the Trisection
Trisection means dividing the line segment into three equal parts. Therefore, we will find two points, P and Q, that divide the segment AB in the ratio 1:2 and 2:1 respectively.

### Step 2: Find the Coordinates of Point P (1:2 Ratio)
Using the section formula, the coordinates of point P that divides AB in the ratio 1:2 can be calculated as follows:

\[ P(x, y, z) = \left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}, \frac{m_1 z_2 + m_2 z_1}{m_1 + m_2} \right) \]

Where:
- \( A(x_1, y_1, z_1) = (2, 1, -3) \)
- \( B(x_2, y_2, z_2) = (5, -8, 3) \)
- \( m_1 = 1 \) (for point P)
- \( m_2 = 2 \)

Calculating the x-coordinate of P:
\[ x_P = \frac{1 \cdot 5 + 2 \cdot 2}{1 + 2} = \frac{5 + 4}{3} = \frac{9}{3} = 3 \]

Calculating the y-coordinate of P:
\[ y_P = \frac{1 \cdot (-8) + 2 \cdot 1}{1 + 2} = \frac{-8 + 2}{3} = \frac{-6}{3} = -2 \]

Calculating the z-coordinate of P:
\[ z_P = \frac{1 \cdot 3 + 2 \cdot (-3)}{1 + 2} = \frac{3 - 6}{3} = \frac{-3}{3} = -1 \]

Thus, the coordinates of point P are:
\[ P(3, -2, -1) \]

### Step 3: Find the Coordinates of Point Q (2:1 Ratio)
Now, we will find the coordinates of point Q that divides AB in the ratio 2:1.

Using the same section formula:
- \( m_1 = 2 \) (for point Q)
- \( m_2 = 1 \)

Calculating the x-coordinate of Q:
\[ x_Q = \frac{2 \cdot 5 + 1 \cdot 2}{2 + 1} = \frac{10 + 2}{3} = \frac{12}{3} = 4 \]

Calculating the y-coordinate of Q:
\[ y_Q = \frac{2 \cdot (-8) + 1 \cdot 1}{2 + 1} = \frac{-16 + 1}{3} = \frac{-15}{3} = -5 \]

Calculating the z-coordinate of Q:
\[ z_Q = \frac{2 \cdot 3 + 1 \cdot (-3)}{2 + 1} = \frac{6 - 3}{3} = \frac{3}{3} = 1 \]

Thus, the coordinates of point Q are:
\[ Q(4, -5, 1) \]

### Final Result
The coordinates of the points that trisect the line segment AB are:
- Point P: (3, -2, -1)
- Point Q: (4, -5, 1)

---