I haven't seen the film Hidden Figures yet, but I have read and listened to some dialogue clips in this NPR Review. In it they mention a Frenet frame and the Gram–Schmidt process for orthonormalization of vectors.
COSTNER: (As Al Harrison) You think you can find the Frenet frame for this data using the Gram-Schmidt...
HENSON: (As Katherine G. Johnson) Orthogonalization algorithm, yes, sir. I prefer it over Euclidean coordinates.
Question: How were Frenet frames and Gram–Schmidt orthonormalization used in spacecraft orbit calculations, and why was this especially important in the Mercury and Apollo eras?
For a reminder of what electronic computers were actually like in the 1960's, below are some screen shots from the official trailer. Floating point ops with Python on a Raspberry Pi 2 is about 10,000 times faster than an IBM 7090 for example.
above: That looks like 100μs per division.