I'm going to go on a trip without my girlfriend.
I see, and what does your girlfriend think about you taking a trip without her?
I told her about it but she thinks that I'm cheating on her.
You want to go on a vacation alone, but I believe you need to include your girlfriend too. How would you feel if she did the same to you and left her home, you would be hurt like she is. Gain more of her respect and invite her along to make her feel like you love her.
No, I won't because she has to realize that I can do what I want.
And what about what she wants? You really need to think about her feelings too if you want to keep dating her.