Given points \( A(4, 1, -2) \) and \( B(6, 2, -3) \). The wire connecting these points is trisected by points \( P_1 \) and \( P_2 \). \( P_1 \) is closer to A and divides the segment in a 1:2 ratio, while \( P_2 \) is closer to B and divides the segment in a 2:1 ratio.
1. Coordinates of \( P_1 \):
Using the section formula for a point dividing a line segment \( (x_1, y_1, z_1) \) to \( (x_2, y_2, z_2) \) in the ratio \( m:n \):
\[ x = \frac{m x_2 + n x_1}{m + n}, \quad y = \frac{m y_2 + n y_1}{m + n}, \quad z = \frac{m z_2 + n z_1}{m + n} \]
For \( P_1 \), \( m=1 \) and \( n=2 \):
\[ x_{P_1} = \frac{1 \cdot 6 + 2 \cdot 4}{1 + 2} = \frac{6 + 8}{3} = \frac{14}{3} \]
\[ y_{P_1} = \frac{1 \cdot 2 + 2 \cdot 1}{1 + 2} = \frac{2 + 2}{3} = \frac{4}{3} \]
\[ z_{P_1} = \frac{1 \cdot (-3) + 2 \cdot (-2)}{1 + 2} = \frac{-3 - 4}{3} = \frac{-7}{3} \]
Thus, \( P_1 = \left( \frac{14}{3}, \frac{4}{3}, \frac{-7}{3} \right) \).
2. Coordinates of \( P_2 \):
For \( P_2 \), \( m=2 \) and \( n=1 \):
\[ x_{P_2} = \frac{2 \cdot 6 + 1 \cdot 4}{2 + 1} = \frac{12 + 4}{3} = \frac{16}{3} \]
\[ y_{P_2} = \frac{2 \cdot 2 + 1 \cdot 1}{2 + 1} = \frac{4 + 1}{3} = \frac{5}{3} \]
\[ z_{P_2} = \frac{2 \cdot (-3) + 1 \cdot (-2)}{2 + 1} = \frac{-6 - 2}{3} = \frac{-8}{3} \]
Thus, \( P_2 = \left( \frac{16}{3}, \frac{5}{3}, \frac{-8}{3} \right) \).
The coordinates of the lantern points are \( P_1 \left( \frac{14}{3}, \frac{4}{3}, \frac{-7}{3} \right) \) and \( P_2 \left( \frac{16}{3}, \frac{5}{3}, \frac{-8}{3} \right) \).