Covariance and contravariance of vectors

Okay, let's imagine that you're playing with two toy cars, one red and one blue. You push them around and notice that sometimes they move in the same direction and sometimes they move in opposite directions.

Now, imagine that these toy cars are like vectors, which are arrows that have both a direction and a magnitude (length). When we talk about covariance and contravariance of vectors, we're talking about how their directions change when we transform, or move, them in some way.

Covariance means that when we transform a vector, its direction changes in the same way as the coordinate system we're using. Let's say we tilt the coordinate system to the right. If we have a vector pointing up in the original coordinate system, it will still be pointing up in the new, tilted coordinate system. The red and blue toy cars will still be moving in the same or opposite directions as before.

Contravariance means that when we transform a vector, its direction changes in the opposite way to the coordinate system we're using. So, if we tilt the coordinate system to the right, a vector pointing up in the original coordinate system will now be pointing to the left in the new, tilted coordinate system. The red and blue toy cars will now be moving in opposite directions from what they were before.

So, in summary, covariance is like the toy cars moving together, and contravariance is like the toy cars moving apart. These concepts are important in math and physics when we need to think about how things change as we move them around in space.