The position vectors of three points A , B and C are a , b and c respectively, where c = ( ^2 ) a + ( ^2 ) b . What is ( a b ) + ( b c ) + ( c a ) equal to?

Step-by-step solution

Given c = ( ^2 ) a + ( ^2 ) b Substituting the value of c in the second term of the expression ( a b ) + ( b c ) + ( c a ) : b c = b (( ^2 ) a + ( ^2 ) b ) b c = ( ^2 )( b a ) + ( ^2 )( b b ) Since b b = 0 and b a = -( a b ) : b c = -( ^2 )( a b ) Substituting the value of c in the third term: c a = (( ^2 ) a + ( ^2 ) b ) a c a = ( ^2 )( a a ) + ( ^2 )( b a ) Since a a = 0 : c a = -( ^2 )( a b ) Adding the three terms together: ( a b ) + ( b c ) + ( c a ) = ( a b ) - ( ^2 )( a b ) - ( ^2 )( a b ) = ( a b ) [1 - ( ^