It is Vijay's ninth half-century in Test cricket and his third successive score of 50-plus after 53 and 99 in the first Test at Adelaide.
India lost Shikhar Dhawan (24) in the morning session while Cheteshwar Pujara (18) and Virat Kohli (19) were dismissed in the post-lunch session.
Pujara was unlucky to be given out caught behind by umpire Ian Gould as the ball had come off his helmet's visor while dodging a short delivery from debutant paceman Josh Hazlewood (2/21).
Vijay was rock solid on the other end, having faced 157 balls. In the post-lunch session, he had Pujara with him in the middle and they looked to increase their 33-run stand for the second wicket.
Vijay brought up his fifty in the first over after the lunch break with a cover drive off Starc. The hosts suffered another injury blow as Mitchell Marsh (1-14) hobbled off with a right hamstring pull.
The 100-run mark came up for India in the 31st over and the duo had added 44 runs until then, looking good for a lot more. But umpire Ian Gould ended Pujara's innings with his decision in the 34th over.
The batsman was stunned by the horrible decision and was almost reluctant to walk back to the dressing room.
Twin-centurion from the first Test, Kohli (19) then walked to the middle. He added 37 runs for the third wicket with Vijay before edging one behind off Hazlewood again.
He squared him up off the first delivery Rahane faced. Then Hazlewood nearly had a return catch as the ball lobbed off his bat but fell just out of reach. Even as Rahane struggled to retain composure, Vijay meanwhile stayed solid at the other end, as India's 150-mark came up in the 50th over.
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