Science

The Physics Philes, lesson 3: A Journey to Vector-ian Times

In which a tutor is procured, lines are drawn, and more time is allotted.

Remember last week when there was so much math? So. Much. Math. Well, for this week at least, we can forget that noise. We’re gonna draw some vectors! (And do a little math. But trust me. It’s not too bad.)

These are vectors:

Vectors! Sorry it’s wrinkled. I sat on it.

Vectors, it is becoming clear, are just super important in physics, so it’s important to know what they are and how they function. A vector is a quantity that has direction and a magnitude. A magnitude is the size of the vector. For example, 10 m/s east is a vector; it has a direction (east) and a magnitude (30 m/s.) Don’t get confused. If you were to say that the car was traveling at 30 m/s, that isn’t a vector because there is no direction. (That is an example of a scalar quantity.)

Pretty simple so far, right? But you know what would be really cool? If we could add vectors together. I will make it so.

Adding vectors isn’t like adding scalar quantities like 3 or 7. When we add vectors what we are trying to come up with is a particle’s (or car’s or person’s) displacement, or distance from the starting location to the ending location.

Let’s add vector A and vector B. (For the purposes of this post, I’m just going to type out the whole word, vector. I know that I’m supposed to use the letter with an arrow over it, but I can’t figure out how to do that. Please leave any tips in the comments.)

I ran out of room on the page after the equal sign.

OK. I know that picture doesn’t make much sense, but here’s the skinny. Let’s say a particle traveled along vector A, then turned and traveled along vector B. How far did the particle actually go from its starting point? One might think that all we would have to do is add the magnitudes of vectors A and B, but one would be wrong.

Look what I did. I stuck the tail of vector B on the head of vector A, then drew a blue vector R from the tail of vector A to the head of vector B. That blue vector is called a vector sum or resultant. Vector R is the distance from the starting point that our little particle traveled. We can also get the direction the particle went.

Neat, huh?

Now, let’s do a sample problem that I totally ripped off from my text book. (It’s not cheating because I totally got it right.)

A hiker hikes 1.00 km north, then 2.00 km east on a flat field. How far and in what direction is the hiker from the starting point?

OK, we’ve got this. Let’s do it together. We need to find the vector sum of two vectors (1.00 km north and 2.00 km east,) as well as the direction the hiker traveled, aka the angle Φ. This is just like the example above. We need to draw this out. Don’t have a pen and paper handy? Don’t worry. I drew it for you.

Look mom! I made a compass and everything!

You’ll notice that I attempted to draw the diagram to scale. That way all I have to do is measure it. The answers I came up with were a displacement vector R about 2.25 km and about a 64 degree angle Φ. But that isn’t very precise. We. Need. Math.

Because the given vectors create a right angle, this should be easy peasey lemon squeezey. All we need is a pinch of algebra and a dash of trigonometry.

You’ll notice that the given vectors create a right angle. Vector R is the hypotenuse of the right triangle. That means all we have to do is use the Pythagorean theorem to find out length:

√(1.00 km)2 + (2.00 km)2 = 2.24 km

To find angle Φ, all we have to do is find the tangent:

tan Φ = opposite side/adjacent side = 2.00 km/1.00 km

Φ = 63.4 degrees

Wow! My manual measurements were pretty close!

I wish I could go into more vector things, but I’m running out of time. (That’ll teach me to leave this post until Sunday evening.) I think vectors are awesome, and I can’t wait to do more work with them.

Before I leave you, I want to send many, many Internet high fives to those who have tried to explain some of this physics stuff to me. Commenter GravitysWings basically wrote a whole post in the comments explaining limits and derivative rules. I don’t think I fully grasp it (in fact, I know I don’t) but I know it will come in handy as I get deeper in. Also, I’d like to thank Carl Tracy of helping me with instantaneous velocity and instantaneous acceleration via a Google Hangout, which is less than idea. What a trooper. Gold stars and high fives!

One more thing. Last week I mentioned that this class is a 12 week course. I thought I could teach myself in that amount of time. I was arrogant. Well, my darlings, I have been humbled. I think that, if I had a teacher, maybe. But I’m teaching myself, so I will likely need more time. But I will finish! That I promise you. What that means for you is that I’ll probably be blogging about physics from now until the end of the world. I hope that’s cool with you. (Also, go thank a teacher. NOW!)

So, physics people of the Teen Skepchick community: Did I get it right? Did I mess up? I know there is a lot I left out. Is there anything I need to clarify?

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Mindy

Mindy

Mindy is an attorney and Managing Editor of Teen Skepchick. She hates the law and loves stars. You can follow her on Twitter and on Google+.

1 Comment

  1. June 26, 2012 at 1:25 am —

    Looks fine to me.

    While I think of it, I hope you’ll forgive another long explanatory comment, but I think that some readers might find this useful in understanding what vectors actually are.

    As an encouragement to stick with it, at the end I’m going to describe a theorem which is probably the most remarkable result in all of mathematical physics. If you know a little physics, and you’ve never heard of Noether’s theorem before, I promise you that by the end of this comment you will understand the central idea behind it, and it may just blow your mind.

    (Disclaimer: For the purpose of this comment, I’m going to assume that space and time are “flat”, that is, Euclidean and Galilean. Things get a little more complicated when you get to relativity, but the ideas are the same.)

    OK, let’s first talk about protractors.

    A protractor is a device for measuring angles. You draw two lines which intersect, line up the protractor appropriately, and you can read off a number (assuming we’re working in degrees) between 0 and 360 which represents the angle between the two lines.

    Now think about a compass. (I mean the rotating magnet device, not the drafting tool for drawing circles.) That is also a device which will give you a number between 0 and 360. This number has the same units as a protractor. But it doesn’t actually measure the same thing as a protractor.

    You use a protractor to measure an angle. You use a compass to measure an orientation. One orientation (pointing towards magnetic north) is chosen as being zero degrees, and the compass lets you tell what other orientations are.

    Now here’s the key point: An angle is the difference between two orientations. A protractor measures angles, but a compass measures orientations. “45 degrees” is an angle, “northeast” is an orientation. “45 degrees” also happens to be the angle (i.e. the difference) between north and northeast.

    If you didn’t get that, here’s another example. Consider the difference between a stopwatch and a clock. A stopwatch tells you the duration of time that elapses between two events. A clock tells you the current time, but to do so requires a “reference time” to which all other times are compared. In the case of a clock, the reference time is even usually different depending on where you live.

    Duration is the difference between two times. “26 Jun 2012 01:42:11 GMT” is a time, “3 hours, 25 minutes, 11 seconds” is a duration, or the difference between two times. A stopwatch measures durations, but a clock measures times.

    (In fact, a clock measures time by measuring duration. That’s why when you first get a clock, you have to “set” it so that it agrees with a reference clock.)

    This concept is all over physics and maths. It’s so simple that you may not ever have noticed it, but it’s one of those things which is obvious once you see it.

    But the concept is so important that it has a fancy name. Mathematicians call it a “torsor”. By the way, I don’t recommend looking up “torsor” on Wikipedia, unless you already know some abstract maths, because the exact definition is highly technical. But the idea behind it is very simple: For many situations, we can only measure difference between “things”, not the “things” themselves.

    By defining a reference “thing” (e.g. north, or midnight GMT), we can talk about “things” as if they were absolute. But the universe doesn’t actually come with a preferred orientation, or a preferred “start time”. We define them entirely for our own convenience.

    OK, now think of points in three-dimensional space and how you’d measure where things are. The universe doesn’t come equipped with a coordinate system. That requires a preferred point (i.e. the origin) and a preferred orientations (e.g. which direction the x axis points in). Once you’ve defined those, you now have a notion of “position”. The “position” of a point is the coordinates of that point, as measured in the coordinate system.

    But, of course, since the universe doesn’t come equipped with a coordinate system, all that you can actually do is measure the difference between positions, or the difference between points.

    That is what a vector is. A “vector” is the difference between two points, or the difference between two locations.

    One of the key points that you should understand about torsors is that you can change the reference system and the torsors don’t change. So, for example, you could pick a direction other than magnetic north to be the reference for your compass. All orientations would change, but angles would still measure the same.

    Similarly, you could pick a different time to be “time zero” (e.g. move to a different time zone) and while times would change, durations would still be the same. And you could pick a different origin for your space, and locations or positions would change, but vectors would still be the same.

    OK, so that’s what a vector is. There are, by the way, plenty of other examples of torsors in physics. One example is energy. There is actually no such thing as an absolute scale of energy (outside of cosmology, anyway). What we measure is the difference between two energy levels, which is a torsor. But we sometimes need a reference, so we often define some energy level to be zero, or perhaps some other quantity if that makes the calculations more convenient.

    Or consider electrical potential. A 9V battery does not produce nine absolute volts at its positive terminal and zero absolute volts at its negative terminal. Rather, the difference in potential between the two terminals is 9V. There is no such thing as absolute electrical potential; all we measure is the difference. But we sometimes need to impose a reference, so we usually define the “ground” (i.e. a conductor driven into the ground) to be zero volts.

    OK. So far, so good. Now this is where it gets really interesting.

    As we noted, the universe doesn’t come equipped with a reference system for any of these important quantities. One upshot is that the physics of some physical system must not depend on the choice of reference system.

    You might, for example, define the instant that you drop a ball as time t=0. But what actually happens when you drop the ball should be the same if you defined t=0 to be at some other time. The universe should not depend on your choice of measurement system.

    Physicists call this a “symmetry”. The physics of the system should be symmetric with respect to where you define t=0 to be.

    But that, in turn, imposes restrictions on what equations can reasonably describe the system. You might actually use equations that aren’t symmetric for convenience, but the laws of the universe had better damn well be symmetric.

    It turns out that these “restrictions” take a very pretty form. They are described by Noether’s theorem, or Noether’s first theorem, named after Emmy Noether who proved it. Emmy Noether, by the way, is universally considered to be the most important woman in the history of mathematics. And her theorem is a fundamental tool of modern physics.

    Noether’s theorem is, of course, technical. But it essentially says this: For every symmetry, there is a corresponding conservation law. There are a couple of qualifications, but it’s essentially that.

    You know those conservation laws, right? Conservation of momentum, conservation of energy and so on? Well, Emmy Nother explained exactly what they are: a conservation law is what you get when a physical system has a symmetry.

    Conservation of momentum, for example. is a consequence of the fact that there is no origin point in the universe. If you shift the entire physics experiment five metres to the left, everything will still work exactly as it did.

    So what about orientation? If you rotated everything 25 degrees clockwise, the experiment should still work. The conserved quantity in this case is angular momentum (which you may not have seen, but probably have heard of).

    And time? It turns out, if you apply Noether’s theorem to time symmetry, the conserved quantity is… total energy!

    There are even more exotic symmetries in physics and they, in turn, give rise to other conservation laws. In quantum mechanics, for example, particles act a bit like waves. Waves have a “phase”, but we can’t measure the phase of those waves directly. We can only measure the difference in phase (sounds familiar?). This, it turns out, results in conservation of electric charge.

    (As an aside, if you know about Heisenberg’s uncertainty principle, you may have noticed that the uncertainties are between pairs of quantities such as position and momentum, or energy and time. Now you know why it’s those pairs of quantities specifically.)

    So when today’s physicists talk about gauge symmetries, spontaneous symmetry breaking, supersymmetry, CPT violation, anomalies… you might not get the details, but this is what they’re talking about and, crucially, this is why it’s important.

    The symmetries that they’re talking about may seem exotic (and some of them are very hard to understand), but they all arise from the fact that you can pick any reference coordinate system, or any of a number of field configurations, or what have you, and all the measurements will turn out the same. Sometimes, these symmetries don’t turn out to be real, and you get some pretty interesting physics as a result.

    But hey, at least you now know what a vector is.

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