Unfortunately, unlike Week 254 and 255, this past week was very unproductive for me. 😔 I only made it through three videos on KA and had to go back and rewatch all three vids multiple times. The concept that was taught in the first video feels very unintuitive to me and so it took me a long time to understand it. The second and third videos talked about concepts that were pretty straightforward but built on what was taught in the first video, so they were also difficult for me to understand. Although it certainly wasn’t my best week, I put in a passable effort, so it could have been worse. That said, I’m definitely hoping I can turn things around and get my momentum back this coming week.

The first video I watched was titled Proof of the Cauchy-Schwarz Inequality. I watched the video from start to finish on Tuesday and thought I had the C.S. inequality completely figured out and then when I watched the next video on Wednesday I realized I had the inequality completely backwards. 🤦🏻‍♂️ Before I get into the screen shots from the video, here’s CGPT’s explanation of the inequality:

My laymen’s understanding of this inequality is that if you have two right triangles where the hypotenuses of each are NOT parallel, then the hypotenuses multiplied by each other will be greater-than the adjacent sides of each triangle multiplied by each other summed with the opposite sides or each triangle multiplied by each other:

• (Hyp₁ * Hyp₂) ≥ (Adj₁ * Adj₂) + (Opp₁ + Opp₂)

This means that the product of the lengths of the vectors will be greater-than dot-product of the absolute value of the vectors. Here are some screen shots of Sal working through the proof for this inequality and then my notes explaining the gist of it and giving two simple examples:

The reason why I struggled with this concept so much was because it seems unintuitive to me that the product of the hypotenuses of two triangles would end up being larger than the product of the adjacent sides plus the product of the opposite sides. On top of the two examples shown above in my notes, I also wrote out a handful of trial-and-error examples on my own to see if it’s true. Turns out it is.

The second video I watched this week, Vector Triangle Inequality, seemed equally confusing in the first half of the video, but then Sal drew what he was talking about using and (x, y) grid and it became completely obvious. Here’s a screen shot of what the diagram I’m referring to and my notes explaining:

As you can hopefully gather, the Vector Triangle Inequality just states that if you have a triangle made out of a vector a plus a vector b, the vector connecting those two vectors, (a + b), will be less-than the length of the sum of the lengths of a and b. (That sounds confusing reading it back but I’m pretty sure what I just wrote is correct.)

Finally, the third video I watched from KA this week was titled Defining the Angle Between Vectors and, once again, the proof of the math was somewhat hard to follow but the final equation seems pretty straightforward. Here’s a screen shot from the vid:

The screen shot above doesn’t show the entire proof, but you may be able to tell that the proof takes the Law of Cosines and applies vectors to it instead of using the lengths of lines. The video works through the proof for the equation but didn’t give any examples so I don’t feel super confident that I’d be able to use it, but I feel like I understand the gist of what’s going on with this equation. 🙃

Finally, to make up for the fact that I didn’t put in enough time on KA this week, on Sunday I watched this video:

As you can see from the title, the video goes through ten things you would want to know about vectors. The first six were things I already knew and felt pretty confident about but the last four were things I still need practice with. They are:

1. Vector equation of a line
2. Equation of a plane
3. Intersection of lines in 3D
4. Intersection of planes

I’m going to rewatch that video at the start of this coming week which will hopefully make understanding vectors and all of the associated forms of notation easier to understand. As a side note, it’s interesting how helpful it is to watch different content creators talk about the same subject I’m studying as they all talk about any given subject in a slightly different way which makes a big difference rounding out my understanding of the given topic.

Like I said at the start of this post, I’m really hoping this upcoming week will be more productive for me and one where I make some decent progress through KA. It’s getting less likely by the week that I’ll be able to get through Linear Algebra, Differential Equations, and Multivariable Calculus by 2026. My goal this week is to get through at least eight videos which would finish off the section I’m currently working through. Hopefully the final eight videos cover concepts that are easier to understand than the Cauchy-Schwarz inequality. 😮‍💨🤞🏼