Tuesday, May 12, 2015

Episode 043: 5 Steps to Break Any Vector Into Components (in Cartesian Coordinates) AND Combine Components Back Into The Original Vector!

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Breaking vectors into components will become habitual before the end of your first semester of physics. You will use it in a variety of places. A few monster examples that come to mind are calculating displacement, finding x and y components of velocity during kinematics problems, almost every force problem requires you to break the forces into components, momentum, torque, the list goes on and on. As this will permeate many waking hours of your physics student life, it would be nice to have a procedure to make the operation a little more painless. Go through these five steps each time to make the process a little more mechanical. It will increase your fluency and increase your accuracy, leaving much less room for mistakes.

1) Identify your coordinate axis and draw it on the paper. This is not always defined in the problem. If they don’t require a particular coordinate system in the problem pick what works best for you. HOT TIP: Try to select an axis that lines up with as many vectors as possible in the problem. This will leave less vectors to break into components!

2) Identify your vector and draw it on the paper. I would advise doing a separate drawing for each vector in the problem. This will avoid a lot of confusion because the drawings will get busy and very messy.

3) Identify your angle and draw it on your paper. This is VERY important. The angle matters because your sin/cos/tan change based on which side is opposite and which is adjacent. You can use whatever you’d like but be detailed with your selection!

4)Use sine and cosine relationships for component definition. This step is essentially a nested “do the 6 sohcahtoa steps” step.

5) Box your answer, you’re done!

Okay, wonderful. So we’ve broken the vector into x and y (usually) components. Wonderful, now what? Well, this is the end of the road for now. This recipe can be used in a variety of situations so we can implement it later. The question is, let’s say we’ve broken several vectors into components, all correctly and meticulously like good students. We perform some vector addition and subtraction, and have our final vector components. Now what? How do we reconstruct our final vector? This is usually what the problem will be looking for anyhow. Well, next we have a 5-step recipe for doing just that.

1) Identify the coordinate axis and draw it on the paper. Sound familiar?

2) Identify your components. These should be handy from doing many operations, or else they were probably given in the initial problem.

3) Use the pythagorean theorem to calculate magnitude. a^2 + b^2 = c^2. Solve for c. Remember that we can consider a vector to be a hypotenuse of a right triangle?

4) Use arctangent to calculate angle, i.e. direction. Remember toa? We’re certainly given the opposite and adjacent side if we’re calculating the hypotenuse, so use those sides to calculate angle. Don’t forget to remember if you are in radians or degrees mode!

5) Box your answer, you’re done!

That’s really all there is to it. As before, to gain fluency, it will be useful to run through some examples. The next episodes will go through 4 detailed examples of doing both of these operations.

Wednesday, May 6, 2015

Episode 042: 5 Critical Vector Properties for Components and the Importance of Coordinate Systems on Vector Operations

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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!