If a , b and c are three non-coplanar vectors and p, q and r are vectors defined by p = a 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

Step-by-step solution

p = b c [ a b c ] , q = c a [ a b c ] , r = a b [ a b c ] Now, ( a + b ) p +( b + q ) q +( c + a ) r aligned = a ( b a ) [ a b ] & + b ( b a ) [ a c ] + b ( c a ) [ a c ] & + c ( c a ) [ a b c ] + c ( a b ) [ a b b ] + a ( a b ) [ a b ] aligned = [ a b c ] [ a b c ] +0+ [ b c a ] [ a b c ] +0+ [ c a b ] [ a b c ] +0 [ scalar triple product is zero if two vectors are same] = [a b c] [ a b c c ] + [a b c ] [ a b c ] + [a b c ] [ a b b ] =3