Wednesday, November 18, 2015

Episode 078: The Devil Is In The Details



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Let’s throw another reason on the already monstrous pile of why most people hate physics. Not only do we have a very non-intuitive mind-bending type of logic which describes our world, but put on top of it, that the nature of our universe is very precise. Objects and particles behave and move with high calculable precision, that’s it’s best and worst quality. Humans are pattern seeking mammals and above all tool users. A long long time ago, our ancestors were able to take advantage of something that has easily made us the dominant force on the planet: the universe is a repeatable place. Patterns exist. Animals migrate and behave according to patterns. Tools are a natural extension of this. If we can understand how something works, it is very easy to either use it as a tool, because the performance is repeatable, or exploit it in a way that is highly reliable. This has come to fruition in recent centuries in the form of manufacturing. We can now produce immense quantities of items by taking advantage of interchangeable parts, physics, a large population which needs similar items to support itself. All of these things are just patterns expressed in different ways.

The ultimate goal in physics is the GUT, or Grand Unified Theory, a.k.a. the Theory of Everything. Potentially the ultimate purpose or use of this species: the grand-daddy of all patterns. As we make our way towards some GUT, if there is one, the patterns we utilize and look for are becoming vastly more complex. Even calculus, which is roughly 400 years old, is something most people can’t even begin to wrap their minds around.

This is the conundrum of our universe: the pattern seeking mammals struggle to see the patterns in it. Physics has a beauty that is very easy to convey to anyone, once of course you remove most of the mathematics. Simply describing how things work: throwing an object on Earth roughly follows the shape of an arch, or parabola. The planets orbit the sun in a circular, or sometimes elliptical orbit. These ideas are fascinating to even children, and children want to know very much why these bodies behave this way. Then you start to explain it. “Well ok, so you have the force of gravity, which is about 9.81 m/s^2, and you have to consider the angle of 42.5 degrees, and the initial velocity of how many m/s?” You start to lose people. Start throwing these values into equations, and it’s over. Unless you’re talking to physics majors, and then it’s not so bad, but you can still lose everyone if you’re not careful. And it’s not that physics majors, or even your average person isn’t smart. Most people are very smart. It’s that the actual patterns that exist around us are very counter-intuitive, and difficult for our jelly-filled brains to take hold of.

On most physics tests, even non-majors can get started off the right way. It’s when we start doing too much too fast, or get careless that most of the points are lost. Any small mistake in the tens or hundreds of individual simple calculations that need to be done can give a wildly incorrect final answer. One decimal point gives you ten or hundred times more or less. One rounding error can throw your final answer off enough to take 20 points off a test. And it’s not just tests. Using the wrong units means that a lander crashes into mars at unbelievable speeds instead of landing like a feather. That is why what we’re doing at this point of the podcast is so important. Learn the fundamentals. Pay attention to the little details. Master these tools. You will have to use them in crazy combinations one day, and there will be an expectation to do it perfectly. It could mean destroying tons of machinery, or an entire letter grade on your final exam. The stakes are high. If you can whittle the problem down into a series of manageable steps, it won’t seem impossible. And, if you can take extreme care, and focus, and think only about the step you’re working on right now, you will be able to take a thousand doable steps to traverse a leap you never thought you could.

Wednesday, November 4, 2015

Episode 077: c x d and d x c, Cross Product Examples Pt. 2





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One thing that we noticed last time, is that

c x d = -(d x c)

Is it possible to prove this? In order to verify, let’s perform the exact same steps with two vectors [vx,vy] and [wx,wy]. First let’s calculate v x w:

Step 1: Identify Two Vectors in Coordinate System as [ax,ay] & [bx,by]- These are given in the problem as [cx,cy] and [dx,dy]. For later reference, let’s identify each component:

ax = cx
ay = cy
bx = dx
by = dy

Step 2: Multiply ax and by- Here we can easily reference step #1 and perform the following operation:

ax * by = cx*dy

Step 3: Multiply ay and bx- Here again we can easily reference step #1 and perform the following operation:

ay * bx = cy*dx

Step 4: Subtract step 2 from step 3, i.e. ax*by-ay*bx:

cx*dy-cy*dx

Step 5: Place a k-hat at the end and box your answer, you’re done! So our answer is:

(cx*dy-cy*dx)k-hat

Next, let’s calculate -(d x c). In order to do this, we’ll simply calculate w x v and then multiply by -1.

Step 1: Identify Two Vectors in Coordinate System as [ax,ay] & [bx,by]- These are given in the problem as [wx,wy] and [vx,vy]. Notice that they are reverse this time. For later reference, let’s identify each component:

ax = dx
ay = dy
bx = cx
by = cy

Step 2: Multiply ax and by- Here we can easily reference step #1 and perform the following operation:

ax * by = dx*cy

Step 3: Multiply ay and bx- Here again we can easily reference step #1 and perform the following operation:

ay * bx = dy*cx

Step 4: Subtract step 2 from step 3, i.e. ax*by-ay*bx:

dx*cy-dy*cx

Step 5: Place a k-hat at the end and box your answer, you’re done! So our answer multiplied by -1 is:

-(dx*cy-dy*cx)k-hat
=(-dx*cy+dy*cx)k-hat
=(cx*dy-cy*dx)k-hat

So, not a full proof, but very convincing nevertheless. It turns out that this is actually true in all cases, in 2-D and all higher dimensions. In order to explore the cross product deeper, we will now perform the same example, but with 3-D vectors, which will have a single new number as our z-coordinate.