If a , b , c are three non-coplanar vectors and p , q , r are defined as p = b c [ a b c ] , q = c a [ a b c ] , r = a b [ a b c ] , then [( a + b ) p + ( b + c ) q + ( c + a ) r ] is equal to

Step-by-step solution

The given vectors p , q , r are the reciprocal vectors of a , b , c . Using the properties of reciprocal vectors, we have: a p = a ( b c ) [ a b c ] = [ a b c ] [ a b c ] = 1 b p = b ( b c ) [ a b c ] = 0 Similarly, for the other dot products: b q = 1 , c q = 0 c r = 1 , a r = 0 Expanding the given expression: ( a + b ) p + ( b + c ) q + ( c + a ) r = ( a p + b p ) + ( b q + c q ) + ( c r + a r ) Substituting the values obtained above: = (1 + 0) + (1 + 0) + (1 + 0) = 1 + 1 + 1 = 3 Answer: 3