If a , b and c are non-zero coplanar vectors, then [2 a - b 3 b - c 4 c - a ] is

Step-by-step solution

a , b , c non-zero coplanar vectors gathered [2 a - b 3 b - c 4 c - a ] =(2 a - b ) [(3 b - c ) (4 c - a )] =(2 a - b ) [12 b c -4 c c -3 b a + c a ] gathered aligned =(2 a - b )[12 b c -3 b a + c a ] &( c c =0) aligned aligned &=24 a ( b c )-6 a ( b a )+2 a ( c a ) & -12 b ( b c )+3 b ( b a )- b ( c a ) &=24[ a b c ]-6[ a b a ]+2[ a c a ] &- -12[ b b c ]+3[ b b a ]-[ b c a ] aligned aligned ( [ a b a ]=& [ a c a ]=[ b b c ]=[ b b a ]=0) =& 24[ a b c ]-[ b c a ] &( [ a b c ]=[ b c a ]) &=24[ a b c ]-[ a b c ] &=23[