In Dragon Ball Super: Broly movie, Vegeta did not use his Super Saiyan Blue Evolution transformation which looks like below.
Why didn't Vegeta use it?
In Dragon Ball Super: Broly movie, Vegeta did not use his Super Saiyan Blue Evolution transformation which looks like below.
Why didn't Vegeta use it?
Well firstly, the form is called Super Saiyan Blue Evolution and it is Vegeta's comparable form to that of Goku's Super Saiyan Blue Kaioken
As for why it wasn't in the movie, there is no definite reason as we don't see any comments from the staff with regard to the same. However, there are several reasons why this could be the case, for instance
In the manga we see Mastered Super Saiyan Blue Goku/Vegeta fighting Broly and it was stated that the movie was a continuation of both the manga and the anime. So yes! There is no definite way to conclude why Vegeta didn't use the transformation apart from speculating based on some of the facts which I mentioned above.
Contrasting Gary's answer, it is wrong. This form was created by Akira Toriyama but it's much different. In the manga, Vegeta did use this form to fight Jiren and it had more power. In the manga, it's much different. It's mentioned by Whis and Berus that Vegeta has evolved. In the manga, this form is called Super Saiyan Blue Evolved. In the manga, there's no X20 Super Saiyan Blue Kaio-ken, but there is something similar that has more power but lasts only a few 10 seconds. It has no name but Goku mentions saying "I'll push to my limit", basically Super Saiyan Blue Limited.
In the Broly movie, when Goku powers up to SSJ Blue, it looks exactly like this form. So Goku did use his strongest powered-up version of SSJ Blue but was not enough for Broly. You have to take note that the movie is like the manga.
As for Vegeta not using his evolved form is quite not answered, but could be it requires more God energy and time to use it which he did not have at that moment. Also, it shows that Vegeta was not able to use his full power. But according to reading a part in the manga, Jaco mentions, "Wow Goku, you're strong", Goku says, "Not really, I fought a Saiyan stronger than me named Broly", Jaco says, "Wow, sounds scary". Later Vegeta answered, "Yeah, but we both easily defeated him". By the sound of this conversation, Vegeta and Goku both using mastered SSJ Blue easily defeated Broly with no fusion. The Broly movie anime exaggerated Broly's power and required fusion which kinda sucks.
I believe it's because Kakarot and Vegeta realised that a power increase of that scale wasn't enough and Broly would just adapt to the new strength. So they needed a greater difference in strength than x20 SSJB/SSJBE.