Question

Let A= Ai I + A2 J + A3 K and B= Bi I + B2 J + B3 K, prove that

A.B = AiBi + A2B2 + A3B3

Answer 1

Multiply out the dot product factors.
(Ai I + A2 J + A3 K).(Bi I + B2 J + B3 K). Cross product terms like AiI.B2J will be zero because the two unit vectors I and J are perpendicular. You will be left with only the sum of the products of components along the same axis, like AiBi.