NDA Mathematics Question (2026) — Solution

Step-by-step solution

Given a , b , c are unit vectors, so | a | = | b | = | c | = 1. Since a is perpendicular to b , a b = 0. The vector c makes an angle of 3 with both a and b , giving c a = 1 1 ( 3 ) = 1 2 and c b = 1 1 ( 3 ) = 1 2 . We are given c = p a + q b + r( a b ). Taking the dot product of c with a : c a = p( a a ) + q( b a ) + r(( a b ) a ) 1 2 = p(1) + q(0) + r(0) p = 1 2 Taking the dot product of c with b : c b = p( a b ) + q( b b ) + r(( a b ) b ) 1 2 = p(0) + q(1) + r(0) q = 1 2 Taking the dot product of c with itself: | c |^2 = p^2| a |^2 + q^2| b |^2 + r^2| a b |^2 Since a and b are orthogonal unit vectors, | a b | = | a || b | ( 2 ) = 1. Substituting the values: 1 = p^2 + q^2 + r^2 1 = ( 1 2 )^2 + ( 1 2 )^2 + r^2 1 = 1 4 + 1 4 + r^2 r^2 = 1 - 1 2 = 1 2 Answer: 1 2