NDA Mathematics Question (2025) — Solution

Step-by-step solution

The given vertices of the rectangle are A(1, 2, -1), B(2, 5, -2) and C(4, 4, -3). The direction ratios of the side AB are (2-1, 5-2, -2-(-1)) = (1, 3, -1). The length of the side AB is 1^2 + 3^2 + (-1)^2 = 1 + 9 + 1 = 11 . The direction ratios of the side BC are (4-2, 4-5, -3-(-2)) = (2, -1, -1). The length of the side BC is 2^2 + (-1)^2 + (-1)^2 = 4 + 1 + 1 = 6 . Now, checking the dot product of vectors AB and BC : AB BC = (1)(2) + (3)(-1) + (-1)(-1) = 2 - 3 + 1 = 0 Since the dot product is zero, AB is perpendicular to BC, which means A, B, and C are consecutive vertices forming a right angle at B. The area of the rectangle is the product of its adjacent sides: Area = AB BC = 11 6 = 66 square units Answer: 66 square units