Let S be the set of all vectors in a two-dimensional space R ^2 . A relation R is defined on S such that for any two vectors a and b in S , a R b if and only if a b = | a || b | . Then the relation R is

Step-by-step solution

For reflexivity, consider a vector a S . a a = | a |^2 = | a || a | Thus, a R a is true for all a S . The relation is reflexive. For symmetry, let a R b . a b = | a || b | b a = | b || a | Thus, b R a is true. The relation is symmetric. For transitivity, let a R b and b R c . Consider the case where b = 0 (the zero vector). For any vector a , a 0 = 0 and | a || 0 | = 0 . Thus, a R 0 is true for all a S . Let a = i and c = j . We have i R 0 and 0 R j . However, i j = 0 , while | i || j | = 1 1 = 1 . Since 0 1 , i is