I will need to read through the rest of the article later, but the initial intuition building is a bit sloppy. None of those vectors drawn in the initial examples belong to the same vector space. Vectors need to emanate from the same origin to be considered as part of the same vector space.
Can you elaborate? Which vector space(s) do these vectors belong to, as drawn?
If anything, this just highlights the sloppiness of these types of illustrations. Things aren't precise enough, in my opinion, for the illustrations to do anything except confuse.
In the mathematical sense, a vector is nothing but an object that can be added to other objects and scaled by some field (with some reasonable properties attached).
In physics, a vector is often more specifically something with magnitude and direction. This still doesn't mean that it needs to be anchored at the origin. Vectors that are anchored at the origin are IIRC called position vectors, but mathematically, if you translate them away from the origin they're still the same vector.
The graph of a function f: X -> Y is the set {(x, f(x)) | x in X}. It is much more clear and precise to associate elements of the graph with vectors such that the 0 vector is identified with the R^2 origin, and then points in R^2 are identified with vectors. Then there is a mapping between vectors in this vector space to the graph, i.e., to points (x, f(x)).
> In physics, a vector is often more specifically something with magnitude and direction.
Physics is sloppy. :) This is not a general description of a vector, where vector is an element of a general vector space. Not all vector spaces have a norm, which is required for magnitude to make any sense.
> but mathematically, if you translate them away from the origin they're still the same vector.
Right, and you cannot always translate vectors without more machinery, such as parallel transport.
> Physics is sloppy. :) This is not a general description of a vector, where vector is an element of a general vector space. Not all vector spaces have a norm, which is required for magnitude to make any sense.
Fine, then let's be precise. An element of the vector space R^n is nothing more than a function from {1,...,n} to R. And every n-dimensional R-vector space is isomorphic to it.
How you wanna draw this in a coordinate system is up to you. It is customary to identify vectors with points on that coordinate system and then equivalently with an arrow pointing from the origin to that point. In that case the zero vector is the origin, or an arrow from the origin to itself.
It is equally customary to use vectors as displacement, i.e. a directed difference between two points. In that case, the vector that you can't actually draw because it has zero length is your zero element. Now your arrows don't have to be anchored at the origin anymore.
Both of these spaces are of course isomorphic.
If your vector space is infinite-dimensional it's of course not gonna be really possible to draw anymore.
Vector spaces have an addition identity normally referred to as the 0 vector.
When drawing vectors, such as on a 2D space, the origin is identified with this 0 vector, and then the points in the 2D space can be associated with the vectors in the vector space.
As if I needed a reason to not read The Atlantic. What in the world is that article even about given the title and subtitle? It's behind a paywall. I can't imagine that the author has zero marine mammal experience, which seems to be a common theme amongst The Atlantic authors (knowing little about that which they write about).
That paragraph you quoted is pretty hyperbolic. Many orcas hunt live and dangerous animals for food. Their prey can seriously injure and even kill them. Because orcas are tight family units with several generations of females and males in the same pod who never leave the pod and because of their intelligence, orcas engage in teaching the younger orcas and each other. So this can very easily look like tortue when it's in fact how orcas train each other to work together. They will also share food readily between each other, so this is why it will also appear to be toying with food.
Yes, there is no doubt that orcas will also legimitaely play with food, but even then, it's a human judgement on a wild animal that can't go pick out meat in a box that comes from an animal raised and slaughtered in a cage.
Humans certainly do kill cats both for food and "fun". And orcas will, from time to time, kill things that they don't eat. Even the exclusively fish eating orcas in the Pacific Northwest have been witnessed killing seals, even though they never eat seals.
Not being afraid of orcas is a show of lack of respect for them. They can weight up to 10,000 pounds, swim at bursts of 35 mph, swim at slower speeds for huge stretches, launch their bodies tens of feet out of the water, and kill anything from salmon and rays to great white sharks, humpback whales, and even blue whales. There is a lot of argument that they may be the greatest predator ever, on land or sea.
A domestic cat can easily wound a human and can usually get away. That is not true of a human in the water and an orca.
Yes, orcas are extremely, extremely intelligent, and there's zero reason for them to eat us for nutrition, as they are highly tuned eaters that maximize their calorie intake per effort. However, a human is completely within an orca's decision process as to whether the human lives or not.
Does anyone have any suggestions for the best way to learn WebGPU/wpgu without much exposure to prior APIs such as OpenGL, WebGL, or Vulkan? As in, I'm wanting to learn WebGPU/wgpu inside and out, but I don't much want to learn about other APIs. But this has proved difficult due to a dearth of resources on WebGPU and it's new shader language, WGSL.
Individuals would have their lives ruined either from massive fines or jail time.
reply