Let A=ia1 + ja2 + ka3
B= ib1 +jb2 =kb3 and
C=ic1 + jc2 +kc3
B.C =b1c1 + b2c2 +b3c3 is a scaler and so
A.(B.C) = (b1c1 + b2c2 +b3c3)(ia1 +ja2 +ka3)
A.(B.C) = ia1b1c1 +ja2b2c2 +ka3b3c3 . . . . . . . . . . .(1)
A.B = a1b1 + a2b2 +a3c3 is a scaler and
(A.B).C = (a1b1 + a2b2 +a3c3)(ic1 + jc2 +kc3)
(A.B).C = ia1b1c1 + ja2b2c2 + ka3b3c3 . . . . . . . . . . (2)
RHS’s of (1) and (2) are identical and so
the LHS’s should be equal and hence it follows that
A.(B.C) = (A.B).C