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According to documentation matrix.ith_to_zero_rotation return a rotation matrix that sends the i-th coordinates of the vector v to zero by doing a rotation with the (i-1)-th coordinate :

sage: v = vector((2,3,1))/sqrt(14)
sage: matrix.ith_to_zero_rotation(v, 2)
[                       1                        0                        0]
[                       0  3/10*sqrt(14)*sqrt(5/7)  1/10*sqrt(14)*sqrt(5/7)]
[                       0 -1/10*sqrt(14)*sqrt(5/7)  3/10*sqrt(14)*sqrt(5/7)]
sage: matrix.ith_to_zero_rotation(v, 2) * v
(1/7*sqrt(14), sqrt(5/7), 0)

See also matrix.vector_on_axis_rotation.