Despite Vegeta's incapacity to wield the Kamehameha, Trunks has displayed total command of the technique (with the exception of his future counterpart). This implies that even in his current state, Vegeta is capable of learning and mastering such a powerful energy wave.
In fact, this power level is more than enough to destroy Earth if used aggressively. However, as we see later on in the series, when used properly with caution, it can also be an extremely effective tool for battle.
Furthermore, since we never actually see Vegeta use Kamehameha, it can only be assumed that he knows how to control it in ways that Trunks could not. Whether this means through mental strength or some other method, we will never know.
However, despite all this evidence, some fans believe that Vegeta cannot perform Kamehameha because he is not truly fit enough to do so. They claim that since we never actually see him use this power, then it must be impossible for him to do so effectively.
But this argument fails because you can't just assume something is impossible until you see it proven otherwise. If we were to deny Goku his victories due to some obscure rule, then we would never learn anything new about the characters.
He wasn't just uninterested in Trunks at first, but Vegeta couldn't be bothered to aid Bulma at all. Vegeta didn't even bother to check on Bulma when she became pregnant. This is the time when Vegeta travelled into space to become a Super Saiyan. When he returned, he found out that she had lost the baby. This makes sense given that it was around this time that Vegeta started training hard for his upcoming fight with Goku.
However, after hearing this, Vegeta's attitude towards her changed. He began to worry about what would happen to her and decided to stay in Earth to take care of her. This made Bulma fall deeply in love with him. She told him so and asked him to marry her. However, due to his overconfidence, Vegeta refused her offer. This makes sense considering that he never planned to die and still thought he could win against Goku.
Later on, after defeating Cell and becoming immortal, Vegeta finally agreed to marry Bulma. But before they could get married, an evil entity named Yamcha came looking for revenge against Vegeta. So he used his mental powers to make Vegeta fall in love with him instead. Then he forced Vegeta to fight each other which ended up killing them both. But before they died, they realized how much they loved one another and decided to remain together forever.
Vegeta's presence in Bulma's life has even inspired him to do better things in his life and on his personal path to success. Vegeta has become an even more strong warrior since he is now attempting to protect his family rather than just beating Goku. He also uses his new found patience and anger management skills to deal with Barazanu who has been stalking the family.
When Bulma first meets Vegeta, she is instantly attracted to him. But because he's a violent criminal, she refuses to have anything to do with him. However, when he turns around and saves her from being killed by Gogeta, she changes her mind and decides to give him a chance. She tells him that if he can control himself around her then they could be good friends. From then on, Vegeta starts going to her restaurant every day where he trains with Goku until it's time for them to fight each other again.
During one of their meetings at the restaurant, Bulma tells Vegeta about how she used to love dancing but then lost her ability after being attacked by a wild boar named Babidi. Since then, she has been trying to find someone who will take care of her business so she can stay home and relax. When Vegeta hears this, he feels sorry for her and wants to find the person who did this to her.
After 7 years of time travel training, Vegeta became Super Saiyan 2. He mastered the ordinary Super Saiyan before obtaining Super Saiyan 2. He was also affected by Babidi's powers, which allowed him to reduce the distance between him and Goku. Vegeta, on the other hand, shattered Babidi's power, which startled Supreme Kai Shin, who believed that no mortal could break Babidi's magic. However, since Vegeta used his full power, it caused him great pain.
Vegeta then told Goku that he would meet him again in 22 hours at Earth's dojo where they will fight for the last time before going back in time. Before leaving, Vegeta promised to train further so that he can become even stronger than he already is.
Meanwhile, on the planet Namek, a furious Frieza attacked Dragon Ball while Android 16 was away on a mission but was defeated by him. Overcome with rage, Frieza decided to kill everyone on Namek except for Pan, who he wanted as his bride. When Android 16 returned, he fought Frieza but was defeated due to being overconfident. After seeing this, Frieza decided to finish off Android 16 but was stopped by Master Roshi, who absorbed the attack from Android 16 using his life force. Roshi then gave Android 16 some tips on how to fight Frieza before dying. Heartbroken, Frieza left Namek alone while searching for another planet where he could continue his quest for revenge.
Back on Earth, Goku and Vegeta met up with Trunks who had just arrived.
Vegeta, on the other hand, was powerful enough to protect Bulma's dignity after Beerus had smacked her. We can't think of many people who could easily overcome Vegeta with his latest Super Saiyan Blue Evolution phase. Bulma has gotten herself one of the greatest protectors the multiverse has to offer by marrying Vegeta.
They were supposed to be married in an Earth wedding ceremony but that didn't happen due to Beerus' attack on it. But still, they did get married in a sense even if it wasn't legal because no law can stop them from getting married. So in conclusion, Vegeta protecting Bulma's honor made them feel comfortable enough to get married in the first place.
Vegeta trained tirelessly after Goku became the first Super Saiyan in years, until he, too, was able to attain the metamorphosis. Once he did, however, his quest for power ended; instead, he used his new-found strength to help others.
He began by saving Krillin from being killed by a monster and later helped save both Gohan and Goten when they were caught in an explosion. Later on, during the Battle of Earth, Vegeta joined forces with Goku to defeat Cell. After this battle was over, Vegeta went back home to Save Town where he restored order and stopped criminals who had gathered there before leaving again for other missions that Goku sent him on.
In conclusion, after Goku transformed into a Super Saiyan, Vegeta trained hard to achieve the same transformation himself. Once he did, however, he decided not to use his power to fight evil, but rather to help those in need. He saved people who were being attacked by monsters and even fought alongside Goku during their final battle with Cell before returning home to continue doing good deeds.