Passage: Let a , b , c be unit vectors. Further, a is perpendicular to b ; c makes an angle 3 with both a and b ; and c =p a +q b +r( a b ) . Question: What is the value of r^2 ?

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