How was Johnson’s policy of increased US involvement in Vietnam part of his liberal vision and how well did his plan work out?
There is much in the way of complexity in terms of reconciling Johnson's liberal vision with the increased US involvement in Vietnam. If one wishes to view him in a tragic sensibility, then one would say that Johnson's liberal vision caused him to plunger deeper into the abyss of Vietnam. In this sensibility, Johnson was almost advocating an internationalized "Great Society," in which he wished to "leave the footprint of America" in Vietnam. Such a notion believes that Johnson wanted to commit to Vietnam in the hopes of Westernizing the nation and liberalizing it so that it would be on par with other nations in the West in terms of schools, hospitals, and providing social servies. The flip side to this would be Johnson's preoccupation with stopping "the domino theory." At some level, one has to concede that Johnson's increased involvement in Vietnam was fostered by the rather Conservative fear of Communism. In this paradigm, Johnson is seen as war hawk, committed to drive out Communism at all costs and not understanding that the Vietnamese were more against the idea of losing their autonomy then they were pro- Communist.
Whether one sees Johnson as a tragic liberal or driven war hawk, Johnson's policies in Vietnam have to be seen as being severely challenged. Not working on reliable intelligence, as well as a flawed premise about the commitment of Viet Cong troops, as well underestimating the resolve of the North Vietnamese and overestimating the strength of the South Vietnamese all contributed to the United States failure in under Johnson.
I would argue, as the first answer does, that LBJ was really acting more as a conservative than as a liberal when he chose to expand US involvement in Vietnam.
However, if I had to argue that his actions were part of a liberal vision, I would say that he increased US involvement in Vietnam to give himself more political leeway to do the liberal things he wanted to do domestically.
LBJ was eager to promote his Great Society programs. But he knew he needed to be strong politically to do this. If he had pulled out of Vietnam, he would have been vilified and would have lost the political clout needed to push his liberal programs through.
So I'd say he sacrificed Vietnam so as to be able to pursue his liberal domestic agenda.