The Deranged Mathematician: What are Vectors?

Many introductory texts (in physics, especially) will define a vector as something that has magnitude and direction. This was a valid definition prior to the end of the 19th century, but mathematicians' concept of what a vector is has considerably expanded since then. Today, it would be far more correct to reverse that statement: something that has magnitude and direction is a vector... but vectors encompass a lot more. One of my favorite ways to demonstrate this is with the card game Set---each card can be usefully (from the pure mathematician's standpoint, anyway) described as a vector, but I struggle to see how any such card can be described as having a magnitude and direction. Incidentally, this example was what motivated me to cover the field of three elements in my linear algebra class---it gave the perfect excuse to play Set with my students while having it be (legitimately!) relevant and educational. And it worked; they still remember it years later. Read the full post (for free) on Substack: What are Vectors?