One of the first and most fundamental operations you'll come across in vector calculus is that of the inner product. It seems simple at first. All you hvae to do is take the components of a vector, multiply those components together, and then add them up. But then, when it comes to interpreting those vectors, most people fall short of understanding it. Yes, it's easy algebraically to define the inner product and you can do so in infinite dimensions. Spoiler alert: it's the i'th component of vector a and the i'th component of vector b multiplied together, then summed up to n dimensions. Yes, that's it that's the inner product. So now you're asking me: Irosa, why are you spending so much time acting like it's complicated? Why are you almost bored with this initial, algebraic interpretation? Isn't this calculation good enough? It told me what to do, I don't need to think too much about it.
Or maybe you don't mind thinking too much. I don't know, I'm not you. But it's frustrating for me, as a thinker, to be tricked into thinking that I'm overthinking. Most people at a certain point realize they don't really understand the inner product and then say that it measures the "similarness" of two vectors. And that's a pretty okay interpretation of the inner-product, but that generates a few very interesting questions that someone should address: using the common interpretation of the inner product, that being as
Or maybe you thought that the "projection" interpretation was correct? That the inner product was the shadow of one arrow onto the real number line? I think that's actually a specific type of inner-product, but in the general case of two vectors, that's not what's actually going on! Perhaps the inner-product is the shadow of the arrow a a distance b into the real number line?! HAHAHAHAHA, hahahahAAHAHAH!!! I'm going crazy here trying to make distances relative to a's and b's, why why why why WHY WHY WHY!!!
Hmnnghghghfghfhghgg... Mhnhghgh...hrrgrggrrggg...mmnn... oh, sorry I was gnawing on a stick there. My brain feels fried! I know you're asking me now: What's the point of all this confusion? Just explain to me what the hell the inner-product is! Irosa just denied me the satisfaction of having the two most common interpretations of the inner-product and now I'm really annoyed!!! So what is YOUR interpretation of the inner-product Irosa?
Well, first, let's all agree that every interpretation of the inner-product serves it's purpose. As an intellectual anarchist, I will inform you that this interpretation I landed upon is only meant for local vs global considerations when dealing with higher-order geometries. You can still use those interpretations when doing something dumber, or stupider... like quantum mechanics! :)
I'm just playing, but the reason I made that joke is that I'm not giving you the correct interpretation or the keys to the universe with this. I'm giving you another, easier conceptual tool when specifically tackling the question of distance. If you're gonna study tensor calculus, this is the version of the inner product that will help you. And this is how I'll word that interpretation:
First, imagine you're in a forest of planes (basically a 2d rectangle projected into a 3d world) and then take a vector a and another vector b. The magnitude of the vector a is the distance you'll be travelling and the magnitude of the vector b is the amount of planes within one meter. The cos (phi) aspect of the inner-product you saw before is just the angle that you walk at; and when those vectors are aligned, you'll walk the maximum amount of distance since you're going straight forward through the planes. If those vectors are perpendicular, you're always walking perpendicular to a plane and thus you never cross any planes when walking. Essentially, the inner-product is a measure of how many planes you'll cross when you walk a distance a through a measure of b things per meter. Essentially, it's a measurement of how far we'd travel if the usual coordinate system were morphed to make b it's unit vectors, and we'd wanna measure our meters relative to that new coordinate system.
Given that interpretation, imagine if you wanted to know how far you'd run if, say, we invented a new measuring system that said one of our new measurement, call it boomba, is defined as the arrow (8,6,4). That's our ruler, our b vector. And say you wanna travel to the point (1 trillion, 6 billion, 500 million). Well, this is our vector a. This interpretation tells us that we'll travel a inner product b meters, which equals (8 trillion, 36 billion, 2 billion). That's how many meters we'll travel in our coordinate system of b!
And you're right! It is really useful! But this interpretation is specific to theoretical physics and differential geometry, because this specific understanding of the inner product helps you understand, conceptually, what Einstein was doing with general relativity. It makes one measurement "relative" to another, and thus allows you to understand the effects of coordinate transformations between vectors. This becomes really important when conceptualizing what's happening in arc length parametrization, which you can read here! However, don't just throw away those other interpretations! Use them! The "shadow" interpretation of the inner-product is more useful for understanding the Gram-Schmidt orthogonalization process, and the "equality" sentiment DOES help you when learning quantum mechanics! Why would you wanna think about relativity and coordinate systems when dealing with quantum systems? Similarity is good enough there, and only in quantum mechanics would you know what you mean by similarity! However, that won't help you in general relativity, so best get to studying buddy!
Just remember, I DO NOT have a teaching degree! I'm merely a self-studier trying my best to communicate how I understand these things. And let's also not forget here, the stated purposes of my essays are FOR ME to better understand the material, not you! If you truly wanna understand this material, you'd be better off doing what I am doing now and forcing yourself to do problems, derive equalities, find other explanations, and most importantly: explaining your interpretation to another person! Writing this explainer helped me see the flaws in my understanding, and because I didn't want YOU to misunderstand what an inner-product was, it helped my mind understand what I was trying to say even better. After this, any flaws in my interpretation are things I couldn't have seen before this moment. And, since I know exactly what I know about this topic, I can improve. If I had never explained this topic to you, I would have never known everything I know about the topic!
There were uknown unknowns in my own explanation, huh... very strange, very weird... So knowledge is like an unknown unknown thing, where there are things you know about what you know and things you don't know about what you know? It's weird. Only when I explain things do I uncover my personal unknown things I didn't realize I didn't fully understand about my own line of thought when it came to inner-products. Like, for example it was so genius how me forcing myself to come up with an example of an inner-product (as good philosophers tend to do when learning a philosophy for the first time) led to that entire 2nd-to-last paragraph above! I'm so smart, wow! Hahaha, lol. I feel so cool and swag now. I'm glad I wrote this. The only reason I even realized that "relativity" thing was that, because I was so on-the-fly with that relativity interpretation that it felt like I was spitting ideas rather than teaching something. I love spitting ideas!