Toss the cabbage, cucumber and carrots together in a heatproof bowl, then set aside.
Bring the vinegars, sugar and salt to a boil in a small saucepan. Pour the vinegar mixture over the cabbage mixture, tossing to make sure all the veggies are covered. Let sit for 15 minutes. Use the slaw to add flavor to your favorite Asian bowl or sandwich. Vegetables can be kept in their pickling liquid in the fridge for up to 2 weeks.
Use a vegetable peeler to cut the carrot into thin strips if needed.