Maybe I'm breaking some forum rules here, but I'd like to respond to the original question: How do you introduce an invented word to the reader?
"Vil" seems to be "technobabble" to me. This term might sound derogatory, but to me, it's a valid means to show your story is set in a future high tech environment. Technobabble has the special property that you might not have to explain it to the reader at all: It's just like mentioning made up names of alien races to show "we are not alone." Maybe the author does not know what it means himself.
Okay, why not just forget about "Show, don't tell"? Checkov (no, not Pavel, Anton) probably didn't mean it this way, anyway. Just tell the reader. Insert a paragraph at the appropriate point and explain to the reader what the hell they are talking about it.
Okay, it's been a long time coming (it was indefinitely delayed by a computer failure (data drive crash), and further offset as I had to check ALL of my files to ensure every change to those files was properly captured), but here's the relatively short passage concerning "vils":
"We're finally past the heliosphere, so we're ready to begin the faster than light phase of our journey."
"Good, I'm eager to get this journey underway." Al concentrated a moment, before waving with his hand. "We need to jog our planned trajectory to port three at positive two by five vils."
"Five vils? That will delays us for some time."
"It can't be helped. After all, you don't want to plunge headlong into something you didn't plan on, would you."
"Wait up," another officer protested. "I haven't achieved the records you have, but I don't sense any impediments to our normal route."
"Excuse me," Al said, sensing a potential problem with others undermining him. "Who are you?"
"Sorry," Captain Yitzl said. "This is Albrechzkl, he's your second."
"I was led to understand there weren't enough Intuits to man the ships you have in the air. How do we rate two?"
Yitzl shrugged. "This has turned into a high priority mission. Beyond that, Albrechzkl isn't a full-time Intuit. He normally works in propulsions, as his natural abilities directed him elsewhere. But it was decided you needed an apprentice, leaning your techniques so we can spread your newer techniques throughout the fleet. Even though he's not up to your standards, and can't approach your prolonged speeds, he'll allow us to travel faster than light even during your off hours."
Al frowned, not having anticipated someone potentially questioning his every move. "Well, Albrechzkl, the conflict isn't an immediate risk of collision. Instead, it's a slight widespread anomaly which will affect the integrity of our journey."
"Okay, now I'm intrigued too," Yitzl said. "What sort of anomaly are we facing, and if there's no risk of collision, then what's the risk we're facing."
Al sighed, pausing before speaking. "That's part of what I was getting at before. Part of why we humans have done so well since arriving, is because of the conflicted nature of your Tandorian aides. We never knew it was unusual, but we have the ability to turn off our aids at will. Not only does this prevent them from countermanding us, it also produces a higher efficiency. Rather than allowing our aids to focus on thousands of different functions, we prioritize them. While we're off duty, we leave most of them running as needed, seeing as we don't know which will be required. But part of my previous performance was because, by shutting down those alternate functions, my aides can detect more detail than other Intuits.
"This anomaly, while risking no widespread calamity, does risk substantial delaying our journey."
"And yet you risk taking us five vils out of our primary path, for a duration of several hours, at a rate of multiple times the speed of light, that'll take us way out of our way. What 'minor inconvenience warrants that much of a disturbance?"
"Instead of a standard obstruction, like colliding with a small comet barreling towards us, I'm detecting a gravitational wave, which will not only affect our overall speed, it's likely to affect our navigation, causing us to accelerate in a new, unanticipated direction—which will require additional time to identify and correct. Thus the diversion, while unfortunate, will be much less severe than we'll face otherwise."
"Are you serious?" Albrechzkl asked. "That's … phenomenal. There have been occasional reports of gravitational wave disturbances, but we've never been able to predict them before."
"Again, it's all a matter of focus. Your aids, while helping a great deal, also constrain the very abilities they augment. Just as they provide additional security, they also rob everyone of the freedom to try new ideas and change the steady progression towards a predetermined destiny."
"Damn, it sounds like that's not something that you can teach others without those innate abilities."
"You'd be surprised," Al said, glancing at Captain Yitzl. "If you don't accept your limitations, you can often accomplish what was previously seen as impossible."
Note: This passage is right off my first draft, so it hasn't been edited or even reviewed, but I wanted to give everyone a view of where the story went with the information everyone volunteered.
As for "vils", I took the 'technobabble' route, not actually explaining it in any detail, but giving a vague idea of what it involves (I explain the role of "Intuits" in space travel elsewhere in the story). The main point of this passage, is that "Al" is trying to buy his team time to act by slowing the spacecraft's travel time, but is stymied by having a second Intuit second guessing his actions. Thus the entire chapter is about the conflict between everyone involved: the humans and the other alien species, the unique role the humans have been given—even if they don't deserve it—and how Al is trying to buy them time to ensure they don't get annihilated when they finally confront the almost certain death they're been assigned to.
The finished product, however, will be much more polished (including many more participants with alternate reactions to the information revealed.