Hello, I'm Doug, stand-up physicist, sitting down to reply to Math the beautiful Pavel Grinfeld, some of the videos he's done saying that tensors are great because they are, but I'm trying to say, you know, I think these space-time numbers, which is a variation on Quaternines, are even better. So in his rules of the game, tensor one video, he writes out the definition of a dot product. So he says, you know, you've got these things that are directed lines and segments, call this one U, and another one called V, and there's this dot product of the two, which is U dot V equals the absolute value of U times the absolute value of V times the cosine of the angle between the two. And that works in any arbitrary dimension, but I am focused on space-time because that's where I get to do E M, the standard model, general relativity, and that's about all I care about being able to do, things are on that kind of physics nature. And I'm exploring what it's like to use this technical variation on Quaternines. So now I see several questions that naturally come up when you think about this. And that is, well, what do you get out of this? Do you get another directed line segment? And of course not, you get a scalar. But what's a scalar's relationship to a vector? They're not the same, but can't you formalize the relationship? And G, you took the cosine, but what about the sine? Shouldn't that be some kind of thing? Well, it is for Quaternines. And this is not a general answer. This won't work in arbitrary dimensional space, but for what we're going to do, there are solid answers to all of those legitimate questions. So we're going to think about a scalar, a combined with that three vector u and a different scalar b and the same three vector v. And say, well, what do you get when you form the space time number product? You get, well, scalar times the scalar is a scalar. And then yes, you do get that exactly the same that product, but there's this minus sign in there. We're not going to worry about it. Because we're not warriors. Cosine, alpha, great. Then, well, for the three vector, how about the scalar times a vector? Well, one of them would be that. And the other one would be u. Let me make sure that's a d u and that one. And then you get the sign over here. And this is now the sign alpha. And that is the product. And now to me, that looks as a feeling of completeness in the sense that this is all the possible ways you can have interact between the scalar and the vector. You got the scalar scalar, you've got the scalar times a vector and the vector times a scalar. And then you have two sorts of products here. You've got a dot product, which they told you about. And you have the cross product. And a lot of people like get crazy about that cross product. Oh, it's so hard. It's so difficult. Well, this way it just looks complete, doesn't it? I mean, you took the sign, the cosine over here. And that was like, yeah, that's easy. It's all the same direction. And now this is kind of the right angle stuff. This is the odd functional kind of expansion of it in a way. So this is u dot v. And this is u cross v. So to me, that is more beautiful. Because it feels complete, there's nothing else you can do between a scalar and a three vector product than that. And that's why I like it. Thank you very much.