The Inspiration4 team blasted off on Wednesday from the Kennedy Space Center in Cape Canaveral atop one of SpaceX's two-stage reusable Falcon 9 rockets.
The Dragon capsule's two men and two women are looking to spend three days circling the world from an unusually high orbit 100 miles (160 kilometers) higher than the International Space Station.
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The mission was part of a fiercely competitive battle between Bezos' Blue Origin and fellow billionaire Richard Branson's Virgin Galactic to tap a potentially lucrative space tourism market.
Born in Kalyan, Maharashtra, Sanjal Gavande is a systems engineer at Blue Origin who always dreamt of designing aerospace rockets.
The nearly 71-year-old Branson and five crewmates from his Virgin Galactic space tourism company reached an altitude of about 88 kilometers over the New Mexico desert.
Sirisha Bandla, a 34-year-old aeronautical engineer, is set to become the third Indian-origin woman to head to space when she flies as part of Virgin Galactic's first fully crewed flight test on Sunday.
While billionaire Branson is set to fly to space aboard his Virgin Galactic's VSS Unity spaceplane on July 11, Amazon Founder Bezos is expected to take off in his own space tourism rocket on July 20.
The winner will be able to claim a seat on the flight with Bezos, his brother Mark, and one other passenger on the flight, which will mark the company's first mission flying humans.
Parts of the Long March 5B re-entered the atmosphere at 10:24 a.m. Beijing time (0224 GMT) and landed at a location with the coordinates of longitude 72.47 degrees east and latitude 2.65 degrees north.