A little bit of an experiment here. Putting up a "rerun" episode. If you knew back then, what you know right now.... A little gauge to see how far you've come in the last 20 episodes. How much more comfortable are you with this topic now? Listen and find out!
When considering how to introduce vectors as an introduction to vector components, I thought that it would be critical to dig into all the nitty gritty of vectors, dot products, cross products, unit vectors, which operations were commutative, etc. I decided that breaking vectors into components is a very useful operation that can be done in very few steps, with a relatively little amount of knowledge. The useful crux here is that you really don't need to have a depth of knowledge to understand what is going on here, be really good at breaking a vector into components, and using this technique all over your physics courses. I'm not saying that the other depth-stuff isn't important, but it's not critical to get the job done here.
1) Vectors can be imagined as an arrow of certain length. The length gives the magnitude, and the orientation of the arrow gives the direction.
2) Vectors slide!- Any vector of the same magnitude and direction is the same as any other in the coordinate system. You can slide vectors anywhere you'd like in the coordinate system and have the exact same properties! E.g. displacement. It doesn't matter where in the world you walk 5 miles, you still walked 5 miles.
3) Vectors can be represented as a single coordinate, [x,y,z]- Because vectors can slide, they can be described as a single point in the coordinate system, with the other point assumed to be the origin.
4) Since vectors can be described in terms of a coordinate, say, [x,y], the vector can be thought of as a hypotenuse, whose legs are the vector components which align with the x and y axis.
5) Vectors can add and subtract by adding and subtracting like components. Many applications, such as displacement, force, actually any vector fare, you will have to add or subtract vectors. The easiest way to do this is to add and subtract the x, y, and z components of each, and then recombine the resulting vector.
Do you see how all of the effort and emphasis on being fluent with sohcahtoa is going to come in handy? Those 6 steps are going to get embedded into a single step in the breaking vectors into components step-by-step. So, we should be applying our newly found sohcahtoa knowledge in no time!