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One way to obtain the complete expression is as follows:

sage: print(' + '.join('({})*({})'.format(a, b) for a, b in zip(u, v)))
(0)*(x) + (1)*(y) + (1)*(z)

~~One way~~ (Edited to ~~obtain the~~ address the follow-up question posted as a comment to an earlier version of this answer).

Obtain a complete expression ~~is as follows:~~using string formatting:

sage: x, y, z = SR.var('x y z')
sage: u = vector([0, 1, 1])
sage: uu = vector([1, 2, -1])
sage: v = vector([x, y, z])
sage: print(' + '.join('({})*({})'.format(a, b) for a, b in zip(u, v)))
(0)*(x) + (1)*(y) + (1)*(z)
sage: print(' + '.join('({})*({})'.format(a, b) for a, b in zip(uu, v)))
(1)*(x) + (2)*(y) + (-1)*(z)

The parentheses are maybe not so nice but they take care of keeping things clean whether coefficients are positive, negative or zero.

If one wants to push the "pretty-printing" effort a little, one should probably write a function. For example:

def axbycz(u):
    r"""
    Return a custom string for the scalar product of
    this vector with the vector (x, y, z), keeping zeros.

    EXAMPLES::

        sage: axbycz(vector([0, 1, 1]))
        '0x + y + z'
        sage: axbycz(vector([1, 2, -1]))
        'x + 2y - z'
        sage: print(axbycz(vector([0, 1, 1])))
        0x + y + z
        sage: print(axbycz(vector([1, 2, -1])))
        x + 2y - z
    """
    aa = lambda c: '' if c == 1 else '-' if c == -1 else c
    bc = lambda c: '+ {}'.format(aa(c)) if c >=0 else '- {}'.format(aa(-c))
    a, b, c = u
    return '{}x {}y {}z'.format(aa(a), bc(b), bc(c))

where the output will be as in the examples included in the function's documentation.

One could instead decide to have the function print a result rather than return a string.

def print_axbycz(u):
    r"""
    Print a custom string for the scalar product of
    this vector with the vector (x, y, z), keeping zeros.

    EXAMPLES::

        sage: print_axbycz(vector([0, 1, 1]))
        0x + y + z
        sage: print_axbycz(vector([1, 2, -1]))
        x + 2y - z
    """
    aa = lambda c: '' if c == 1 else '-' if c == -1 else c
    bc = lambda c: '+ {}'.format(aa(c)) if c >=0 else '- {}'.format(aa(-c))
    a, b, c = u
    print('{}x {}y {}z'.format(aa(a), bc(b), bc(c)))