Final answer:
Plasmid vectors that did not integrate a foreign DNA fragment will replicate independently in the host cell without providing any new genetic information intended from the cloning process.
Step-by-step explanation:
Plasmid vectors that did not integrate a foreign DNA fragment in their polylinker regions will replicate independently in the host cell. Since they do not contain the foreign DNA, they will not disrupt any of the genes such as the lacZ gene.
These plasmids, therefore, will continue to replicate as usual within the host bacterium, such as Escherichia coli, and convey any existing traits they have, like antibiotic resistance. However, without the integration of the foreign DNA, such plasmids will not provide the host cell with any new genetic information that was intended to be introduced through the cloning process.