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Let's Invent Two-Dimensional Multiplication

Wrath of Math

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An intuitive derivation of the geometric definition of the dot product, a multiplicative operation between two elements of the cartesian plane R^2. We'll consider how multiplication of real numbers works, and how the many directions between vectors in R^2 can be translated into a meaningful factor in our operation. We'll then see the component definition of the dot product, and prove it is equivalent to the geometric one using the law of cosines. We'll complete several examples and finish by discussing angles and dot products in four dimensions. #mathematics #multivariablecalculus #maths

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0:00 Intro to 2D Space
1:10 Our Objectives
1:27 Creating the Dot Product
10:17 The Cosine Function
14:07 Defintion
14:53 Radians
15:47 Trying the Dot Product
18:23 An Easier Way
20:43 Vector Operations
22:35 Law of Cosines
24:19 Self Dot
25:32 Proof of Equivalence
29:01 Finding Angles
31:20 Angle in 4D
33:54 Conclusion

20 Comments

@WrathofMath Creator 1 year ago (edited) Pinned

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@kent631420 1 year ago

bro discovered the dot product 🔥🔥🔥

@glassman96 1 year ago

Why is the math saying lul at me stop laughing at me math

@angeldude101 1 year ago (edited)

Multiplying 2D numbers is easy: (ax + by)(cx+ dy) = acx² + adxy + bcyx + bdy². The problem is that the result we got is 4D, but we'd probably want the result to stay in 2D. For starters, let's define x as 1, because a multiplicative identity is useful, which gives us ac + (ad + bc)y + bdy². That shaved off a dimension of the result, but we still ended up with a trinomial rather than a binomial, so we just have to pick something for y² to be. Since scalar factors are mostly irrelevant, that really only leaves 5 options: y² ∈ { 0, 1, -1, y, -y }. Choosing y² = -1 gives the well-known complex numbers. Choosing y² = 1 gives the less well-known hyperbolic numbers (more often called the worse name of "split-complex numbers"). Choosing y² = 0 gives the dual numbers. I'm actually not sure what would happen when choosing y² = ±y, but it's possible that it'd actually just result in a shifted or skewed copy of one of the prior three.

@calebdupree3564 1 year ago

i turned on the video thinking “wouldn’t it just be like multiplying matrices?” and then you started talking and i was like “i guess not” and then you kept talking and i was like “it is just like multiplying matrices”

@notthecheatr23 1 year ago

This is important stuff but where it really gets beautiful (and starts to seem much more systematic) is when you get to geometric algebra... perhaps that one's considered a higher barrier to entry, but it does a great job of tying together a lot of "weirdness" like dot and cross products, complex numbers and generalizing to higher dimensions, etc.

@viniciusares321 1 year ago

Good dot product explanation but, a real 2D multiplication should result in a 2D result, and everything be still contained in a 2D plane without third dimension

@Dani0x1B 1 year ago

Freya Holmer made a video about vector multiplication a while ago that's very well explain and I recommend it to anyone who enjoyed this one

@douglasmagowan4918 1 year ago

Alternatively, you can define two dimensional multiplication such that we multiply the magnitudes and add the arguements. This is isomorphic to the multiplication of the complex numbers, and is a natural extension of multiplication of the real numbers.

@xbirdshorts5075 1 year ago

Fun fact: to multiply numbers in the complex plane, you can construct the result through convering the incoming numbers into polar system, constructing the modulus as the multiplication of the incoming numbers, and theta as the sum of thetas. For example, -2+2i (r is 2sqrt2, theta is 3pi/4) by 1/sqrt2+i*/sqrt2 (r is 1, theta is pi/4) is -2sqrt2 (r is 2sqrt2, theta is pi)

@nicholasleclerc1583 6 months ago

Basically the "Geometric Product", or an introduction to the intuition & the "main idea" (")behind(")

@AjCohn 1 year ago

This is so awesome. I'd love a followup about matrix multiplication.

@Blox-Beam 1 year ago

3:34 thats basically what modulus of a complex number is

@slushpupp1 1 year ago

6:25 OH YEAH, VECTOR!!!

@BKNeifert 6 months ago (edited)

6:45 Yes, because you're creating a different shape other than a square. I love math, because all the principles work together, and build on one another, so what's simple is true, but you also can add layers of nuance to it, and completely change the idea.

@kingofgoldness 1 year ago

I thought this was gonna go into how imaginary number multiplication adds angles 😥 cool video about vectors still

@bele13 6 months ago

20:40 Isn't it generally better to differentiate between scalars and vectors in their representation? Then, you'll know what the dot means because you'll know what types of objects it's working on.

@kodirovsshik 1 year ago

No, it was exactly what I thought.

@josip_anton_bilic 1 year ago

This is the only video that made me fully understand the dot product and the use of it.

Could you by any chance make a video about cross products in a similar fashion to this one as i dont quite get it either?

@elitettelbach4247 1 year ago

I really like your handwriting 🥰
The various colors of sharpies explaining mathematical concepts takes me back to my days of watching vihart when I was in school ❤️