I had an absolute terrible start to this week but managed to claw it back on Saturday and Sunday and ended up achieving my goal of watching all the remaining eight videos in the Vector Dot- and Cross-Product section. That said, I didn’t make notes on the final two videos I watched and found them very confusing. So, I’m planning to start next week by watching those two videos again and trying to make sense of them. The other six videos I watched were generally pretty easy for me to follow along with but I definitely found parts of them confusing. All in all, I can tell I made some decent progress better understanding linear algebra this week, in general, and how to use the cross product, specifically. I think I said this last week, but after having gotten rocked in Multivariable Calculus for so long not knowing how/why the linear algebra worked, it’s a relief to finally be making a bit more sense of it. 😮‍💨

The first video I watched was titled Defining a Plane in R3 with a Point and Normal Vector. I think my notes did a decent job of explaining what the video talked about (although they don’t explain the proof that Sal did in the video) so, without going into detail, here are a few screen shots from that vid and my notes:

The second video I watched, Cross Product Introduction, was the easiest video for me to understand of all of them. In it, Sal just went through the basics of what the cross-product is and how to use it, i.e. the step-by-step process of taking the cross-product of two vectors. The way he worked through it was a method I’ve seen before but not the typical way I do it, so I made a few notes on the difference between the two methods. Here’s a screen shot from that vid and then my notes:

The last picture of the notes I took was just me stating that a dot-product can be taken of any two vectors in any dimension but that a cross-product of two vectors can ONLY be taken in R³. This is something I might have heard in the past but it never stuck with me. It was helpful hearing Sal talk about this as it helped me with the visualization of what’s going on with the cross-product, and for some reason helped me to remember that a dot-product outputs a scaler value and also the method of how to take the dot-product.

The screen shot just below came from a video titled Proof: Relationship Between Cross-Product and Sin of Angle which ended up being somewhat followable (which apparently is a word 🤔) but was also pretty intense (as you can see from the screen shots below). I didn’t write out any notes that are worthy of posting here, but I did make a note to mention that: 

1. The angle between the dot-product of two vectors is the product of the length of both vectors and the COSINE of the angle between them, and 
2. The product of the length of both vectors and the SINE of the angle between them equals the absolute value of the cross-cross product of the two vectors.

The fifth video I watched this week was titled Vector Triple Product Expansion (Very Optional) and it was relatively easy to follow but somewhat confusing to understand what was being taught. I’m glad I watched it but I’m not going to bother explaining it here (partly because I would have a hard time if I tried and might not be able to), but here’s a couple of screen shots from that vid and my note summarizing the formula it left me with:

Finally, the sixth, seventh, and eighth videos I watched this week were all related but difficult to understand. I somewhat understood the first video of the three but only on a surface level. The video taught me that the formula for a plane floating in a 3D space is Ax + By + Cz = D, and that a normal vector coming off a given plane is essentially the same formula, n ⃗ = ax + by + cz. Honestly, I don’t really understand how it works and definitely don’t understand why it’s true. In fact, I could be completely wrong with what I just wrote and it might not actually be true… But that’s how I understand it at this point. Anyways, here’s a screen shot I took from the sixth video which I’m pretty sure if a proof for a normal vector coming off a plane having essentially the same formula as the plane itself:

And that’s going to do it for this week’s post. Once again, not my greatest pot but, considering how little I’d accomplished halfway through the week, I’m happy that I had anything to write about at all. As a side-note that’s somewhat related, there were a handful of crappy things that happened to me this week which slowed me down working on KA and I’m pretty sure this coming week won’t be as bad, so my fingers are crossed I can have a better week and make some decent progress on KA. (What else is new?) The silver lining of all of this is that I’m at least understanding what I’m learning more easily than what I was learning in Multivariable Calculus. I’m praying to the math gods (but also all the gods) that things will get a bit easier for me and that I can get back to crushing KA. 🙏🏼 😤 💪🏼