A and C are the points (1, 3, -2) and (4, -3, 4) respectively - Scottish Highers Maths - Question 11 - 2016

To find the coordinates of point B that divides the line segment AC in the ratio 1:2, we can use the section formula.

Let A be represented by the coordinates (1, 3, -2) and C by (4, -3, 4). The section formula states that the coordinates of point B, which divides the line segment joining points A(x1, y1, z1) and C(x2, y2, z2) in the ratio m:n are given by:

$B(x, y, z) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right)$

In our case, m = 1 and n = 2:

1. Calculate x-coordinate of B: $x = \frac{1 \cdot 4 + 2 \cdot 1}{1 + 2} = \frac{4 + 2}{3} = \frac{6}{3} = 2$

2. Calculate y-coordinate of B: $y = \frac{1 \cdot (-3) + 2 \cdot 3}{1 + 2} = \frac{-3 + 6}{3} = \frac{3}{3} = 1$

3. Calculate z-coordinate of B: $z = \frac{1 \cdot 4 + 2 \cdot (-2)}{1 + 2} = \frac{4 - 4}{3} = \frac{0}{3} = 0$

Thus, the coordinates of point B are (2, 1, 0).