Prove A.(B*C)=(A*B).C​

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

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