def cross(a, b, spatial_dim, output_dtype=None):
    """xarray-compatible cross product
    
    Compatible with dask, parallelization uses a.dtype as output_dtype
    """
    # TODO find spatial dim default by looking for unique 3(or 2)-valued dim?
    for d in (a, b):
        if spatial_dim not in d.dims:
            raise ValueError('dimension {} not in {}'.format(spatial_dim, d))
        if d.sizes[spatial_dim] != 3:  #TODO handle 2-valued cases
            raise ValueError('dimension {} has not length 3 in {}'.format(d))
        
    if output_dtype is None: 
        output_dtype = a.dtype  # TODO some better way to determine default?
    c = xr.apply_ufunc(np.cross, a, b,
                       input_core_dims=[[spatial_dim], [spatial_dim]], 
                       output_core_dims=[[spatial_dim]], 
                       dask='parallelized', output_dtypes=[output_dtype]
                      )
    return c

import numpy as np
import xarray as xr
a = xr.DataArray(np.empty((10, 3)), dims=['line', 'cartesian'])
b = xr.full_like(a, 1)
c = cross(a, b, 'cartesian')