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 use the section formula:

Let the coordinates of A be ( A(1, 3, -2) ) and those of C be ( C(4, -3, 4) ). The formula for finding the coordinates of point B is:

$B = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}, \frac{m z_2 + n z_1}{m+n} \right)$

where ( m ) and ( n ) are the ratios in which B divides the segment (in this case, ( m = 1 ) and ( n = 2 )).

Calculating each coordinate:

1. For the x-coordinate: $B_x = \frac{1 \cdot 4 + 2 \cdot 1}{1 + 2} = \frac{4 + 2}{3} = \frac{6}{3} = 2$

2. For the y-coordinate: $B_y = \frac{1 \cdot (-3) + 2 \cdot 3}{1 + 2} = \frac{-3 + 6}{3} = \frac{3}{3} = 1$

3. For the z-coordinate: $B_z = \frac{1 \cdot 4 + 2 \cdot (-2)}{1 + 2} = \frac{4 - 4}{3} = \frac{0}{3} = 0$

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