UPSC NDA I 2025 Mathematics Paper

Given,
Points A(1,2,−1),B(2,5,−2), and C(4,4,−3) are three vertices of the rectangle.
We need to find the area of the rectangle formed by the vectors AB and BC .
Length of vector AB :
Length of AB=√‌(2−1)2+(5−2)2+(−2−(−1))2
=√‌12+32+12=√‌11
Length of vector BC:
Length of BC=√‌(4−2)2+(4−5)2+(−3−(−2))2
=√‌22+(−1)2+(−1)2=√‌6
Area of the rectangle is the product of the lengths of vectors AB and BC :
Area =√‌11×√‌6=√‌66
∴ The area of the rectangle is √‌66 square units.