I was close to reaching my goal for this week which was to get 50% of the way through the unit Vectors but came up a bit short. I finished reviewing the unit Trigonometry on Thursday and got 40% of the way through Vectors by the end of Saturday. Working through the remaining two sections in Trigonometry, which dealt with more trig identities, was much more difficult than I thought it was going to be. Although I feel like I got better at using the identities (which seem to essentially be formulas), I still don’t completely understand how they work. It’s the type of thing where I can’t visualize in my head how and why each component of the formula identity effects the solution. Regardless, I’m going to leave them for now and will hopefully wrap my head around them when they inevitably come up further into calculus.

One key thing that became clear to me while reviewing trig was that sin(θ) is always a y-coordinate and cos(θ) is always an x-coordinate, and that both values will have an absolute value >0 and <1 since they’ll both necessarily be less than the the hypotenuse which always equals 1. (This concept was explained to me over and over again when I initially went through trig but for whatever reason it didn’t stick until this week.) Understanding this concept helped me to understand that a^2 + b^2 = c^2 is that same thing as sin^2(θ) + cos^2(θ) = 1^2:

The identities I don’t understand are angle sum, difference, and doubling identities:

What confuses me, for example, is adding two angles that each have a y-coordinate > 0.5. For example, if sin(θ) and sin(Φ) are 0.99 and 0.98 respectively and each are inside the unit circle with their respective hypotenuses equaling 1,  I understand that you’re “stacking” the two triangles on top of each other (i.e. adding them together) but, by doing that, the hypotenuse of their sum would then become > 1 which doesn’t make sense to me. I think what must be happening is that you don’t factor in the hypotenuse of both triangles initially, just the angle of each triangle. I’m not sure, however, and it’s frustrating not being able to visualize what’s going on. Furthermore, there are also identities KA referred to as “Reciprocal and Quotient” identities, “Half Angle” identities, and “Cofunction” identities all of which seem even more confusing than the sum, difference, and doubling identities which I also don’t understand. 😡

When I moved on to the unit Vectors I was happy to realize the content wasn’t too difficult and that it shouldn’t take me too long to get through even though it’s fairly big (1400 M.P.). From what I’ve learned so far, it sounds like vectors are an important part of physics which is a subject I’m very interest in learning about so, again, this was another thing I was glad to realize. The very first thing I learned in this unit was the difference between a vector and a scaler:

As you can see from the photo, the main difference between the two is that a vector describes how far an object moves AND the direction the object moves (a.k.a. the objects displacement) whereas a scalar ONLY describes how far the object moves (a.k.a. the distance).

The majority of the two days I worked on this unit I spent learning the definitions and notation used with vectors:

As you can see, there are a few ways to write vectors. You can write two capital letters with an arrow above them (ex. AB^→) or a single letter with an arrow above it (ex. a^→). In the second and third photo you can see that the vector coordinates can be written horizontally like standard x-, y-coordinates but, as you see in the last photo, they can also be written vertically with square brackets with the x-coordinate being written above y-coordinate. As shown in the fourth photo, another thing worth noting is the magnitude of a vector (a.k.a. the hypotenuse of the vector on an x-, y-coordinate plane) is denoted with two bars on either side of the letter(s) used to denote the vector (ex. ||a^→|| or ||AB^→||).

Lastly, I worked on adding and subtracting vectors as well as multiplying a vector by a scaler:

As you can see from the first photo, when adding and subtracting vectors to and from each other you simply add/subtract the x-coordinates and y-coordinates. For example:

• a^→ = (3, 2)
• b^→ = (7, -5)
  
  • a^→ + b^→ = (3 + 7, 2 + (-5))
    • = (10, -3)
      
  • a^→ – b^→ = (3 – 7, 2 – (-5))
    • = (-4, 7)

Multiplying vectors by a scaler is just as straightforward. As you see in the second photo, you simply use the FOIL method to multiply each coordinate by the scalar.

I’m looking forward to working through Vectors (560/1400 M.P.) since it doesn’t seem too difficult right now and plus the fact that Sal mentioned that vectors play a role in physics. If I’m able to get through the remainder of the unit at the same pace, I should be able to get through it by the end of the week and hopefully start the following unit, Matrices (0/1700 M.P.). Once I get through the ladder unit, I will be ~80% of the way through the course which will give me a shot at finishing the course by the end of the month. Considering I was initially aiming to get through the course by the end of January, I’m REALLY hoping I can finish it off by the end of February. I’ve been working for 75 weeks to get started on calculus and I’m don’t think I can wait much longer!