(Mainly asking for suggestions) Question about what courses to take +

Algebra Level 2

Hey! So, I'm not really an expert in science and math, science is I guess a passion of mine but I guess that's for all of us, but when I started using brilliant, I looked at the courses there were, and I chose astronomy because of course, that's a topic that makes us dream, and, well, I learned a huge amount of concepts and learned a lot, understanding was not once a problem in astronomy, but since I wanted to know all the concepts to maybe use them, I memorized all the formulas etc, and that was really the only challenge, BUT

After I was done with the astronomy course, I wanted to keep learning about things like it, so I went on to 'quantum objects'. Now, thing is when I reached about the 2nd chapter, a lot of things became confusing, with variables and concepts I had a hard time getting, and since I was understanding less and less, but didn't want to just get the gist of it, I looked at what courses brilliant suggests taking before quantum objects, and linear algebra showed up.

Now, I assumed it was pretty simple for the most part, I always noticed in science that, obviously science is linked to math, but although the more advanced the science is, the more complex the math that comes with it usually is, I tend to see that the math needed to understand science tends to be much easier than the actual science, like I'm not sure, I feel like 9th grade science would involve basic 3rd grade math, 10th grade science stuff, 5th grade, not sure, something like that, but pretty much, I imagined it would be somewhat easy.

Now, when I saw 'linear algebra', I thought it might be a somewhat deeper look into multivariable equations, maybe some systems of equations and a few new principles, because I saw it listed under advanced math, but when I started the course, it started with 2 and 3 unknown systems of equations, and immediately went to matrices, then applications, and, well, so you can understand my current level of math: I'm in 11th grade currently on algebra 2, did algebra 1 last year and I get most algebra 2 concepts (otherwise my grades might not be too great), I have never seen calculus, I'm not supposed to be taking the course in school before I graduate high school, and so far, I'm struggling with linear algebra but with rereading, taking notes and sometimes watching videos to help understand everything, I'm so far getting it (about 2/5 chapters in).

Basically, my main question is: based on the fact I'm trying to understand quantum mechanics, I'm taking the linear algebra course, I'm in algebra 2 right now in school and have never heard of calculus other than 'it's a math course', what do you suggest I do? (keep going with linear algebra then do calc done right, do another course before even going to calculus done right because otherwise I won't get it either, or take a course before linear algebra but skip calc, etc idk) I don't want to take many courses because I don't have all that much time, so as long as I can understand the most of linear algebra then understand quantum mechanics clearly, I guess I want to take as little extra as possible. Thanks!

Also, it might be great to also see a list of what courses build up to what I'm trying to learn, because I know that there is a 'prerequisites' and 'next steps' place where you find what courses are suggested next, but a course tends to include another course as a next course, which also features that first course as a next course, so it's somewhat confusing and since brilliant suggests that calculus done right is suggested before linear algebra, I imagine it might be easier? Not sure, well thanks again! Also, I'm not sure if there is a section for questions, all I found was questions and answers, so I guess I will make an algebra question:

PROBLEM: Given the following: vector v which passes through the origin also goes through [x=2, y=-3, z=1] vector w which passes through the origin also goes through [x=-3, y=4, z=5]

What is the angle between the 2 given vectors? Firstly, we can write an expression which will involve this: a⋅b=∥a∥∥b∥cosθ where cosθ is the angle between the 2 vectors, a represents the values of the first vector's given cords, b for the 2nd vector.

You can express this as a= [2, -3, 1] You can express b as b= [-3, 4, 5] a b = (a1 b1 + a2 b2 + a3 b3) or 2 -3 + -3 4 + 1*5 . ||a|| means absolute value, or every number's positive value - distance from 0. You can find absolute value by rootsquaring the square of a number, since any number square is positive, so, since a=[2, -3, 1] and b=[-3, 4, 5], ||a|| means sqr(2^2 + -3^2 + 1^2), and the same applies for b, but you put b values instead of a values.

Finally, to find the degree instead of cosθ, you can do arcos(cosθ) to find θ.

What is θ?

13/70 0.491 180 60.57

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1 solution

Robin Minecraft
Apr 10, 2019

So, you are looking for cosθ, so you want to firstly as demonstrated substitute a b into a1b1+a2b2...anbn (where n is 3, only 3 dimensions given), so, a b becomes a1b1+a2b2+a3b3 = (2 -3)+(-3 4)+(1 5)=-13 so you know a b=-13 now, we need ||a|| and ||b|| in: -13 = ||a|| ||b|| cosθ. ||a|| = sqr(a^2), and since the matrice of a is 2, -3, 1, ||a||=sqr(2^2+(-3^2)+1^2) = sqr(4+9+1) = sqr(14). ||b|| = sqr(b^2), so, same thing applies: matrice of b is -3, 4, 5, so ||b||=sqr(9+16+25) = sqr(50)

So, we know that a b=-13, ||a|| = sqr(14), ||b|| = sqr(50). We can now place: -13 = sqr(14) sqr(50)*cosθ so, cosθ = -(13sqr(7))/70 θ = Arcos(-(13sqr(7))/70) = 119.4296.

Not sure how 60.57 became the 'right answer' but I don't know where to change that, but if a*b had been 13 instead of -13, that would have been the answer, both could be possible as they have a cosθ <= 1 and a degree of 0 to 360, but the real answer should be 119.4296.

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