If b and c are unit vectors and | a |=7 , a ( b c )+ b ( c a )= 1 2 a , then angle between the vectors a and c and angle between the vectors b and c are respectively.

Step-by-step solution

Given: a ( b c ) + b ( c a ) = 1 2 a , | a | = 7 , and b , c are unit vectors. Using the vector triple product identity x ( y z ) = ( x z ) y - ( x y ) z : a ( b c ) = ( a c ) b - ( a b ) c b ( c a ) = ( b a ) c - ( b c ) a Substituting into the original equation: ( a c ) b - ( a b ) c + ( b a ) c - ( b c ) a = 1 2 a Noting a b = b a , the c terms cancel: ( a c ) b - ( b c ) a = 1 2 a Rearranging: ( a c ) b = ( 1 2 + b c ) a Since a and b are generally non-parallel, both coefficients must be zero: a c = 0 and 1 2 +