In this novel, cars (actually it's not the cars but the way people drive them) are a symbol of the attitudes of the rich. Practically everyone in the story is a bad driver, especially the rich people. They do not seem to care about anything -- their lives are empty. This is symbolized by their driving.
In Chapter 3, the main incident with a car is the one where the car Owl Eyes is riding in crashes in the ditch near Gatsby's home. The driver is so drunk that he does not realize he can't drive away since the car has lost a wheel.
The other incident occurs in the end of Chapter 3 with Jordan Baker (the famous, and likely rich athlete) driving Nick (middle class Middle American) around.
she passed so close to some workmen that our fender flicked a button on one man's coat.
Their conversation continues and Nick encourages her to be more careful with her driving. She asserts that she doesn't have to because other people will get out of her way.
Unfortunately, this is a sad commentary on the rich and their attitude toward everyone else. Every situation in which they drive, they go to fast and end up hurting others. This is indicative of the 1920s. Everything was too fast, and it ended in serious hurt, but not just for the poor, for everyone.
The first incident in chapter 3 is the one in which Owl Eyes, one of Gatsby's guests, crashed his vehicle into a ditch, losing a wheel in the process. When Nick Carraway asked him about the incident, he replied,
“I know nothing whatever about mechanics”
“I know very little about driving — next to nothing. It happened, and that’s all I know.”
He was not, however, the one who had driven the vehicle. The driver was clearly drunk and had no idea what happened. He believed they had run out of gas and could not even understand the extent of the damage, believing that the car could be reversed and filled up at a gas station. Against all advice, he still insisted on trying. His incredulity was the result of a drunken stupor.
The second incident relates to an occasion in which Nick was accompanied by Jordan Baker. He commented that she was a "rotten driver" because she passed so close to men working on the road that the car's fender "flicked a button on one man’s coat."
Jordan insisted that she wasn't a bad driver at all; when Nick suggested that she should be more careful or not drive at all, she claimed that she was careful. Nick insisted that she was not, and she retorted that others were.
Cars are symbols of wealth and privilege, since only those who have the means can afford them. The two incidents clearly illustrate this because those who attended Gatsby's parties were, supposedly, the upper crust of society. Jordan Baker was definitely of means, since she was a famous golfer and kept company with wealthy people.
The fact that both the first driver and Jordan displayed such an off-handed manner symbolizes their carelessness and indifference. They were more than prepared to display their good fortune, but were not committed to take responsibility. Their vehicles were mere objects, and they were careless with them. What happened in these two instances epitomizes this attitude. Clearly, their approach was that others were there to take responsibility, as Jordan so pertinently declared.
It is sad that this approach is also reflected in the characters' relationships with people, who they also treat as objects. They have a shallow superficiality about themselves, and what they present to the world is a picture of robustness and privilege but, all in all, their lives are empty and, essentially, meaningless.