BUG: `vdot` of object arrays along zero-length is `None`

Describe the issue:

Following the bug discussed in #31019 for vecdot, np.vdot appears to show similar behavior for zero-length object arrays: it returns None instead of a zero-like scalar result.

For comparison, with the dtype=float, the result is 0.0.

Reproduce the code example:

import numpy as np

cases = [
    np.empty((0,), dtype=object),
    np.empty((1,0), dtype=object),
    np.empty((0,1), dtype=object),
    np.empty((0,0), dtype=object),
]

for x in cases:
    print(x.shape, np.vdot(x, x))

Error message:

Output: 

(0,) None
(1, 0) None
(0, 1) None
(0, 0) None

Python and NumPy Versions:

2.6.0.dev0+git20260623.4357ad1
3.12.3 (main, Mar 23 2026, 19:04:32) [GCC 13.3.0]

How does this issue affect you or how did you find it:

This issue was identified by an automated testing tool currently being developed by our research team.
Before reporting it, we reviewed the code, documentation and checked the latest development version of Numpy to confirm this observation.

Run the supplied examples against the current NumPy development version and trace the np.vdot entry point, using the related vecdot bug (#31019) as context. Check the zero-length object-array path and add regression coverage for the four shapes; done means vdot returns a zero-like scalar rather than None, consistent with the float case.