Parsing sentences to create a chat bot
From a personal standpoint, I think this method is fine. You might want to re-work it a little. I'd recommend creating an array with possible intersections, and looping through that array and checking user input against those. For example:

var intersections = [
    ["what", "do", "you", "think", "about"],
    ["what", "is", "your"],
    ...
];


Finally, I'd recommend having something that allows for multiple keywords, rather than one. In addition to this, you may not want to be checking the length of intersections based on a constant, in this case, three.

Anyways, if there's anything else that you want me to cover, just mention it in the comments, and I'll see what I can do.