Section Formula In Three Dimensions

The technique is the three-dimensional section formula: a point dividing the join of (x_1, y_1, z_1) and (x_2, y_2, z_2) internally in ratio m:n has coordinates ((m x_2 + n x_1)/(m+n), (m y_2 + n y_1)/(m+n), (m z_2 + n z_1)/(m+n)). This weighted-average formula is a core JEE Advanced result. With m:n = 1:3, m + n = 4, and endpoints (2, -1, 3) and (4, 3, -1): x = (1\cdot4 + 3\cdot2)/4 = (4 + 6)/4 = 10/4 = 5/2; y = (1\cdot3 + 3\cdot(-1))/4 = (3 - 3)/4 = 0; z = (1\cdot(-1) + 3\cdot3)/4 = (-1 + 9)/4 = 8/4 = 2. So the point is (5/2, 0, 2). Option (3, 1, 1) is the midpoint, ratio 1:1. Option (7/2, 2, 0) uses ratio 3:1, swapping m and n. Option (2/3, -1/3, 1) divides by the wrong total. This applies the internal section-formula theorem. Plausibility check: the point lies between the two endpoints since each coordinate falls within the corresponding endpoint range, consistent with internal division.