I can see the uses for finding the midpoint of two points, but there is an arrow drawn to m. Are there any needs for vector m?
keenan
The arrows are just meant to suggest that points in the plane can also be viewed as vectors relative to the origin. But are there reasons to average two vectors rather than two points? Sure—a simple but fun example is the so-called "boids" model of flocking. Here's a video. Here, one needs to average velocity vectors with other members of the flock. Another very different kind of example is Nesterov's heavy ball method for optimization, where one (roughly speaking) averages the current gradient vector with the previous descent direction to get a more efficient descent direction. And on and on...
I can see the uses for finding the midpoint of two points, but there is an arrow drawn to m. Are there any needs for vector m?
The arrows are just meant to suggest that points in the plane can also be viewed as vectors relative to the origin. But are there reasons to average two vectors rather than two points? Sure—a simple but fun example is the so-called "boids" model of flocking. Here's a video. Here, one needs to average velocity vectors with other members of the flock. Another very different kind of example is Nesterov's heavy ball method for optimization, where one (roughly speaking) averages the current gradient vector with the previous descent direction to get a more efficient descent direction. And on and on...