Praise be to Allaah.
If a person swears an oath to commit sin, sever ties or cause mischief, it is haraam for him to fulfil this oath, rather he must break the oath, and he has to offer expiation, which is feeding or clothing ten poor persons or freeing a slave. If he cannot do any of these things, then he must fast for three days.
The Prophet (peace and blessings of Allaah be upon him) said: “By Allaah, if Allaah wills, I do not swear an oath then see something better than that, but I expiate my oath and do that which is better” Narrated by al-Bukhaari (3133) and Muslim (1649).
So you should offer expiation for your oath, and let your mother ride with you in the car.
We also remind you to fear Allaah with regard to your mother, for Allaah has mentioned her rights alongside His, so do not refuse any request she makes, or disappoint her, or make her angry with regard to any matter, great or small. Rather praise Allaah for honouring you with the opportunity to honour her and earn her love and approval, for Paradise lies in you staying close to her feet, and taking care of her, looking after her and treating her kindly.
We ask Allaah to help us all to do good and to accept our repentance.