For the specific example you’ve given: dividing an (l,m,n) array by (m,) you can use np.newaxis:
a = np.arange(1,61, dtype=float).reshape((3,4,5)) # Create a 3d array
a.shape # (3,4,5)
b = np.array([1.0, 2.0, 3.0, 4.0]) # Create a 1-d array
b.shape # (4,)
a / b # Gives a ValueError
a / b[:, np.newaxis] # The result you want
You can read all about the broadcasting rules here. You can also use newaxis more than once if required. (e.g. to divide a shape (3,4,5,6) array by a shape (3,5) array).
From my understanding of the docs, using newaxis + broadcasting avoids also any unecessary array copying.
Indexing, newaxis etc are described more fully here now. (Documentation reorganised since this answer first posted).