Question

A man needs to hang two lanterns on a straight wire whose end points have coordinates A (4, 1, -2) and B (6, 2, -3). Find the coordinates of the points where he hangs the lanterns such that these points trisect the wire AB.

Hint:

Quick Tip: To trisect a line segment, use the section formula. The formula allows you to divide a line in a given ratio, which is useful when dividing a segment into equal parts.

Answer 1

We are given the coordinates of points A and B as: \[ A(4, 1, -2), \quad B(6, 2, -3) \] The total length of the wire is the distance between points A and B, and the points where the lanterns are hung divide the wire into three equal parts. Therefore, we need to find the points that trisect the wire. Let the coordinates of the points that trisect the wire be denoted as \( P_1 \) and \( P_2 \), such that: \[ P_1 \text{ divides the wire in a ratio of } 1:2 \quad \text{(closer to A)} \] and \[ P_2 \text{ divides the wire in a ratio of } 2:1 \quad \text{(closer to B)} \] 1. Finding the Coordinates of \( P_1 \): To find the coordinates of \( P_1 \), we use the section formula. The section formula states that the coordinates of a point dividing a line segment in the ratio \( m:n \) are given by: \[ x = \frac{m x_2 + n x_1}{m + n}, \quad y = \frac{m y_2 + n y_1}{m + n}, \quad z = \frac{m z_2 + n z_1}{m + n} \] Here, we are dividing the line segment \( AB \) in the ratio \( 1:2 \), so \( m = 1 \) and \( n = 2 \). Applying the section formula: \[ x_1 = \frac{1 \cdot 6 + 2 \cdot 4}{1 + 2} = \frac{6 + 8}{3} = \frac{14}{3} \] \[ y_1 = \frac{1 \cdot 2 + 2 \cdot 1}{1 + 2} = \frac{2 + 2}{3} = \frac{4}{3} \] \[ z_1 = \frac{1 \cdot (-3) + 2 \cdot (-2)}{1 + 2} = \frac{-3 - 4}{3} = \frac{-7}{3} \] So, the coordinates of \( P_1 \) are: \[ P_1 \left( \frac{14}{3}, \frac{4}{3}, \frac{-7}{3} \right) \] 2. Finding the Coordinates of \( P_2 \): Similarly, to find the coordinates of \( P_2 \), we divide the line segment \( AB \) in the ratio \( 2:1 \). Using the section formula with \( m = 2 \) and \( n = 1 \): \[ x_2 = \frac{2 \cdot 6 + 1 \cdot 4}{2 + 1} = \frac{12 + 4}{3} = \frac{16}{3} \] \[ y_2 = \frac{2 \cdot 2 + 1 \cdot 1}{2 + 1} = \frac{4 + 1}{3} = \frac{5}{3} \] \[ z_2 = \frac{2 \cdot (-3) + 1 \cdot (-2)}{2 + 1} = \frac{-6 - 2}{3} = \frac{-8}{3} \] So, the coordinates of \( P_2 \) are: \[ P_2 \left( \frac{16}{3}, \frac{5}{3}, \frac{-8}{3} \right) \] Thus, the coordinates of the points where the lanterns are hung are \( P_1 \left( \frac{14}{3}, \frac{4}{3}, \frac{-7}{3} \right) \) and \( P_2 \left( \frac{16}{3}, \frac{5}{3}, \frac{-8}{3} \right) \).