The LaTeX feature I installed here is no longer working, so I'm moving to Wordpress, where LaTeX support is built in. The new location for the blog is
http://arcsecond.wordpress.com/
Sunday, October 5, 2008
Monday, September 29, 2008
Re-examing the Inner Product (Euclidean Space)
I'm auditing Kip Thorne's "Applications of Classical Physics" this year. Today was the first day of class, and Kip gave a preamble which, if I may be allowed to paraphrase in a wild, inaccurate, and completely unfair manner, (which is basically to say every word of this is existed only in my head) went something approximately like this:
Awesome.
One thing that interested me was our classes' definition of the inner product between two vectors. Since we wanted to do everything in a frame-independent manner, we couldn't simply define the inner product by
because that presumes the existence of basis vectors. The other common definition is
which is good, but we wanted something that could generalize to the Minkowski space of special relativity. So the definition we came in two parts. First, we defined the inner product of a vector with itself. Then use that definition to bridge to the inner product of arbitrary vectors.
For a vector
, define
where
is the square of the physical invariant interval between two events. To measure this interval, have an unaccelerated clock move from the origin to the event. (
should be real, meaning the square of the interval is positive, and 
is negative. So the inner product of the vector between timelike events with itself is negative). In the case of spacelike events, you should instead find the inertial reference frame in which the events are simultaneous, and lay down a measuring stick between them to get the interval.
Now we know how to take the inner product of a vector with itself. Define the inner product of two vectors and 
by
This is more subtle than it might initially appear. You can't just go willy-nilly with the algebra and start simplifying that right hand side out. You don't know any properties of this inner product, so you cannot, for example, write
Instead you have to work with the actual sums. It was claimed in class that this definition of the inner product is bilinear in the arguments, which wasn't obvious to me. So I asked about it, and Kip suggested I try to work it out for myself in Euclidean space using the Pythagorean theorem. So here goes.
Break down the vector into a component parallel to and a component perpendicular to
The vector
now becomes 
, where 
and 
are unit vectors in the directions parallel and perpendicular to .
Because these two vectors make a right angle, the Pythagorean theorem applies.
similarly,
This is great, because the right hand sides of those equations are just numbers. Expand that out, subtract the two equations, and divide by four to obtain
That's just the normal definition of the dot product, and is obviously linear in both vectors. Now, how about Minkowski space?
Welcome to class. Here is the website. Please call me "Kip". I'm 68 years old and my signature lucky ponytail is a thing of the past. There's not enough time left for ridiculous formalities. I am about to retire. I'm going to start writing books and making movies and various other things that famous retired physicists can do. I'm too old to waterski, so this is the next best option. This is will be last class I ever teach. There will be no grades. I would prefer you actually learn some stuff. Now, let's suppose the laws of physics are frame-independent, and see what restrictions this places on force law dealing with a classical field in Minkowski space..."
Awesome.
One thing that interested me was our classes' definition of the inner product between two vectors. Since we wanted to do everything in a frame-independent manner, we couldn't simply define the inner product by
because that presumes the existence of basis vectors. The other common definition is
which is good, but we wanted something that could generalize to the Minkowski space of special relativity. So the definition we came in two parts. First, we defined the inner product of a vector with itself. Then use that definition to bridge to the inner product of arbitrary vectors.
For a vector
where
Now we know how to take the inner product of a vector with itself. Define the inner product of two vectors
This is more subtle than it might initially appear. You can't just go willy-nilly with the algebra and start simplifying that right hand side out. You don't know any properties of this inner product, so you cannot, for example, write
Instead you have to work with the actual sums. It was claimed in class that this definition of the inner product is bilinear in the arguments, which wasn't obvious to me. So I asked about it, and Kip suggested I try to work it out for myself in Euclidean space using the Pythagorean theorem. So here goes.
Break down the vector
The vector
Because these two vectors make a right angle, the Pythagorean theorem applies.
similarly,
This is great, because the right hand sides of those equations are just numbers. Expand that out, subtract the two equations, and divide by four to obtain
That's just the normal definition of the dot product, and is obviously linear in both vectors. Now, how about Minkowski space?
Why Am I Posting This Crap?
Working with physical models inevitably leads to mathematical problems. In physics education, many professors build up enough tools and techniques to shove these problems aside, and then quickly move on to the real physics. That's simply a reflection of the fact that they care deeply about the physics itself, and I appreciate their efforts to keep students from getting bogged down in details. But taken to the extreme, this attitude produces calculating-maniacs with tons of knowledge and little mathematical intuition. Take at look at the canonical book on math methods for physics (Arfken), and you'll see that it's simply an encyclopedic dictionary of methods.
I'm only now becoming mature enough to go back and re-examine many of the things I learned earlier with an eye towards appreciation rather than computation. This practice intrigues me. The purpose of a blog is ostensibly to create content for public consumption. That's not what I'm doing. I'm working through things for myself. Not because I feel a dire need to understand. Because I can get enraptured by the absurd power of a little solid thought to unearth something I never saw before, and yet later becomes laughably obvious.
According to Google Analytics, in the last month, 58 total people have visited the site in 172 separate visits, and viewed 268 pages. They've spent a total of 5 hours, 41 minutes reading the posts. That's only a small fraction of the cumulative time it took me to write the posts. Also, a lot of the statistics I quoted are just me, logging in from various computers, trying to crank up the scores so I'll feel more important.
Nikita, Julia, and Kangway have actually been participating and giving me feedback, which is pretty much incredible, so thank you for doing that. But looking at the recent posts I've written, I realized a few things:
Today I finished a post about vector products and determinants. I was aware of the geometric interpretations from a few sources, but I had never put the pieces together for myself. By the end, I was amazed. Not only had I built up the idea of a determinant, motivated geometrically, but I had also shown how to use the determinant to solve a system of equations. I didn't even know that would pop out of the argument. If you had asked me to demonstrate how a determinant works before I started writing, I probably wouldn't even have known. Additionally, the path that led me there was remarkably short, and since I had worked through it myself, it walks through a logical (to my mind) problem-solving process.
Math books tend to define and develop the determinant as a number characterizing a linear operator on an n-dimensional vector space. The treatment is more abstract. The proofs showing how to calculate it and how it solves systems of equations frequently rely on manipulating sums of indices and leave me feeling unsatisfied. I believe them, and see that they're correct, but I have difficulty visualizing them as well as the author of the book. I don't knock their approach. I think it's probably the best way. But this supplementary cogitation on the blog has bolstered my appreciation.
Physics books, on the other hand, will normally skip over the entire journey and roll the answer out right away, then tell you to start inverting some matrices. Save the math questions for your linear algebra class!
Writing this blog is my way of taking control of my own educational process. I think about whatever I want, at my own pace, in my own way. Why then post it on the internet? Anything I write has probably been written somewhere else, and probably with more insight and clarity than I'm providing. If my goals are personal education, why paste it all over this blog, advertising it for people to see?
If I were to keep these developments to myself, I wouldn't think them through fully. I'd work until I saw the gist of the argument, then say to myself, "the rest is just mopping up details from here. better move on to something more important."
If I don't take the time to mop up the details, the bits of understanding I did build flit away. So by assuming the responsibility to clean things up to the point where someone could potentially read through the post and work out the results for themself, I force myself to clean up the ideas in my own head. That's what I'm after.
I'm only now becoming mature enough to go back and re-examine many of the things I learned earlier with an eye towards appreciation rather than computation. This practice intrigues me. The purpose of a blog is ostensibly to create content for public consumption. That's not what I'm doing. I'm working through things for myself. Not because I feel a dire need to understand. Because I can get enraptured by the absurd power of a little solid thought to unearth something I never saw before, and yet later becomes laughably obvious.
According to Google Analytics, in the last month, 58 total people have visited the site in 172 separate visits, and viewed 268 pages. They've spent a total of 5 hours, 41 minutes reading the posts. That's only a small fraction of the cumulative time it took me to write the posts. Also, a lot of the statistics I quoted are just me, logging in from various computers, trying to crank up the scores so I'll feel more important.
Nikita, Julia, and Kangway have actually been participating and giving me feedback, which is pretty much incredible, so thank you for doing that. But looking at the recent posts I've written, I realized a few things:
- not that many people care about things such as the geometric interpretation of a vector product rule
- people who do care were better students the first time around than I was, and mostly already know
- people who would care, but don't already know this stuff, most likely don't have the assumed background to learn much from my posts.
- even if the content of the posts is "just right" for you, I skip a bunch of steps and gloss over the parts of the arguments that are less interesting to me, because they're too tedious to draw/
for the blog. Hence, following the posts from scratch would be a rough road. It's not the sort of reading people are looking for when browsing through blog posts. They want quick, to-the-point, easily-comprehensible and entertaining content. This blog is the antithesis.
Today I finished a post about vector products and determinants. I was aware of the geometric interpretations from a few sources, but I had never put the pieces together for myself. By the end, I was amazed. Not only had I built up the idea of a determinant, motivated geometrically, but I had also shown how to use the determinant to solve a system of equations. I didn't even know that would pop out of the argument. If you had asked me to demonstrate how a determinant works before I started writing, I probably wouldn't even have known. Additionally, the path that led me there was remarkably short, and since I had worked through it myself, it walks through a logical (to my mind) problem-solving process.
Math books tend to define and develop the determinant as a number characterizing a linear operator on an n-dimensional vector space. The treatment is more abstract. The proofs showing how to calculate it and how it solves systems of equations frequently rely on manipulating sums of indices and leave me feeling unsatisfied. I believe them, and see that they're correct, but I have difficulty visualizing them as well as the author of the book. I don't knock their approach. I think it's probably the best way. But this supplementary cogitation on the blog has bolstered my appreciation.
Physics books, on the other hand, will normally skip over the entire journey and roll the answer out right away, then tell you to start inverting some matrices. Save the math questions for your linear algebra class!
Writing this blog is my way of taking control of my own educational process. I think about whatever I want, at my own pace, in my own way. Why then post it on the internet? Anything I write has probably been written somewhere else, and probably with more insight and clarity than I'm providing. If my goals are personal education, why paste it all over this blog, advertising it for people to see?
If I were to keep these developments to myself, I wouldn't think them through fully. I'd work until I saw the gist of the argument, then say to myself, "the rest is just mopping up details from here. better move on to something more important."
If I don't take the time to mop up the details, the bits of understanding I did build flit away. So by assuming the responsibility to clean things up to the point where someone could potentially read through the post and work out the results for themself, I force myself to clean up the ideas in my own head. That's what I'm after.
What does a determinant have to do with a cross product? pt. 2
In the previous post, I showed that the quantity
and the determinant
are both tests that the three vectors , , and 
are independent. Now we'll go about showing that they're actually the same test - that they're the same number.
You may be familiar with the geometric interpretation of as the volume of a parallelipiped (box) with sides , , and .
Just a quick explanation -
is a vector perpendicular to both, whose length is the area of the parallelogram they form. See stolen image below:
If I take that and dot it with , I get the projection of into the perpendicular direction, times the area of the box. That is, I get the height of the box times the area of its base. That's the volume. Wikipedia visualization:
There's one subtlety, which is that the volume of the box records its "handedness" in the sign of. By handedness, I mean that if we looked at a mirror image of the box, its volume would be multiplied by negative one. If you do that funny thing with your right hand, and find that points the same sort of direction that does, you have a righthanded box. Otherwise it's left-handed.
This interpretation of as the volume turns out to be the same as the determinant. To see why, we'll make a list of properties that the area of the box has, and then show that those properties imply the normal calculation of the determinant.
and similarly for and
Imagine we have two 2-D vectors in component form, and we put them in a matrix. Then we ask for the determinant. The first property says the determinant of the identity is one.
Using linearity in the vectors, we find
Now we switch the vectors in the identity and multiply the determinant by -1.
and
Finally use the linearity within a vector, which states
where we could drop the determinants of matrices with an entire column zero because they are dependent vectors.
That is the determinant for a 2-by-2 matrix. If you want to do it for 3-by-3, you can use the exact same process of breaking it down into 27 matrices with three nonzero entries each, and then see by permutations of the identity which ones are zero, and where the plus and minus signs go. Out will fall the normal expression for a 3-by-3 determinant.
So that's where that comes from.
, 
, 
such that
where, , , and 
are all arbitary vectors.
We worked far enough just to test if the problem had a solution, but we found this result:
From there, we need to find
by dotting it with a unit vector perpendicular to both and .
Plug this into the equation for, and the stuff about 
cancels, leaving
Those are determinants. That's why they matter. They let you solve systems of linear equations, such as the system of linear equations that results from the problem we just solved.
and the determinant
are both tests that the three vectors
You may be familiar with the geometric interpretation of
Just a quick explanation -
If I take that and dot it with
There's one subtlety, which is that the volume of the box records its "handedness" in the sign of
This interpretation of
- If the sides of the box are orthonormal (all perpendicular to each other, and all unit length), its volume is one (or minus one for a left-handed box).
- The volume of the box is linear in a vector. By this I mean that if we multiplied any side of the box by some constant, we'd multiply the volume of the entire box by that constant. Also, if we make two boxes that share two vectors but differ in the third, the sum of the volumes is the same as the volume of the box whose third side is the sum of the vectors. Notationally:
and similarly for
- If you switch the roles of any two vectors, you'll multiply the volume of the box by minus one. This is obvious if you flip
Imagine we have two 2-D vectors in component form, and we put them in a matrix. Then we ask for the determinant. The first property says the determinant of the identity is one.
Using linearity in the vectors, we find
Now we switch the vectors in the identity and multiply the determinant by -1.
and
Finally use the linearity within a vector, which states
where we could drop the determinants of matrices with an entire column zero because they are dependent vectors.
That is the determinant for a 2-by-2 matrix. If you want to do it for 3-by-3, you can use the exact same process of breaking it down into 27 matrices with three nonzero entries each, and then see by permutations of the identity which ones are zero, and where the plus and minus signs go. Out will fall the normal expression for a 3-by-3 determinant.
So that's where that comes from.
Why it Matters
In the previous post, we were trying to solve the problem of taking three vectors and finding a linear combination of them that yielded a fourth vector. That is, findwhere
We worked far enough just to test if the problem had a solution, but we found this result:
From there, we need to find
Plug this into the equation for
Those are determinants. That's why they matter. They let you solve systems of linear equations, such as the system of linear equations that results from the problem we just solved.
Saturday, September 27, 2008
Debate Debate
Relying Exclusively of Volatile Eruptions of the Reptile Brain Mark: You should watch the debate.
Preposterously Idealized Self SErving Rational Mark: Will it be interesting?
REVERB Mark: You better be interested! It's the Presidential Debate!
PISSER Mark: I don't follow that.
REVERB Mark: You're a functioning, adult member of the society now. You live in a democracy, and that's a great privilege. It's your responsibility to take part in the governance of this nation, and you didn't even vote in the last election! Now you better turn on that debate.
PISSER Mark: Given my demographic, occupation, community, and personal interests it's not difficult to guess which candidate I support. I doubt that watching the debate will change that.
REVERB Mark: That's not the point. Even if you know who you're going to vote for, you still need to stay well informed.
PISSER Mark: Why?
REVERB Mark: Because you're more privileged than any but the tiniest fraction of human beings alive, and yet you take it all for granted. You live in a nation where you personally can have real, meaningful participation in the political process, and it's your responsibility to exercise that right.
PISSER Mark: I admit I frequently take the perks of life in America for granted, but I don't see why following politics is a responsibility, rather than a personal choice. I'd rather do something else.
REVERB Mark: What do you think politics is, then?
PISSER Mark: A hobby.
REVERB Mark: A what?
PISSER Mark: A hobby. A pastime. Like being a sports fan. You can read about sports in the newspaper, listen to experts talk about it on the radio. Then you can choose which team you like, and "support" them by buying bumper stickers and paying for admission to their games. All the same applies with politics, except the bumper sticker becomes a sign on your front lawn and the ticket to the game becomes a vote. They're both just ways for people to have something to get riled up about without doing much harm.
REVERB Mark: Not an apt analogy at all. Sports are just games. It doesn't really matter who wins the World Series. But it sure matters who wins the Presidency!
PISSER Mark: A little bit, maybe. My job prospects might be a little dimmer with one candidate in charge than the other. Their differences on social issues could affect how pissed off the people around me are. The foreign policy they enact will cause me to read different headlines in the news.
REVERB Mark: That's not just news items. All that stuff is really happening out there, to real people. It's not just all about you. To some people it makes a huge difference who's President.
PISSER Mark: But we're discussing my personal decision on whether to watch the debate.
REVERB Mark: Look, I know what you're thinking. You're thinking, "my vote is just one gumball in Candy Land. Doesn't make a bit of difference. The vote in my state always goes to the same party, and one more vote cast either way can't change the result." But you know what? One hundred thousand votes can change the result. And what if one hundred thousand people just like you are using the same logic?
PISSER Mark: Are you saying that if I go to vote it will change one hundred thousand lazy people's minds?
REVERB Mark: You know that's not what I'm saying. I'm saying you shouldn't vote because you think your particular vote is going to make all the difference. You should vote because it's a statement that you give a rat's ass about the future of this country! Not just for yourself, but for everyone you share this country with. And since the US is a leading world superpower, everyone you share this Earth with. And before you vote, you need to be informed. So you need to watch the debate.
PISSER Mark: Watching just one debate won't make me informed. Becoming a competent participant in democracy is a significant commitment.
REVERB Mark: One that every adult American ought to make.
PISSER Mark: Have you met many adult Americans? On the whole, they're not the ones I want making any important decisions. Basically, ability to choose a good Presidential candidate will undoubtedly fall on a curve like this:
So let the people who are innately interested in the whole process, and can do a good job of it, pour their efforts into politics.
REVERB Mark: All the more reason for you personally to get involved. You're way at the top of that curve. Remember high school? You don't want to common man making the decisions - then YOU better get involved and do something about it.
PISSER Mark: That's not really my point. I don't want to invest huge amounts of time and energy into becoming an "informed citizen", which by the way makes the dubious presupposition that it's possible to make a good choice from behind a computer screen, because it doesn't interest me much. Even if I did watch the debate, I don't know enough about foreign policy or economics to decide who I agree with. I'd inevitably fall back on judging demeanor and rhetoric. I could do more good by putting that effort into something I'll enjoy.
REVERB Mark: Like what? What can you do that's more important than your civic duty?
PISSER Mark: I don't know... plant a tree or something, I guess.
REVERB Mark: Oh come on. You've never planted a tree in your life.
PISSER Mark: I did, actually. I planted the free pine tree seedling they gave me third grade on Arbor Day. I think dad accidentally ran over it with the lawn mower.
REVERB Mark: Not the point. That's not what you will do with that extra time. What you will do is read stupid websites or play flash games, or something inane like that.
PISSER Mark: Or maybe I'll study for a while. Or work on that project I had planned, or...
REVERB Mark: Oh please. You've already got three tabs open and they're Pacman, SomethingAwful.com, and then something more awful than that.
PISSER Mark: sigh. Where is it streaming from?
Preposterously Idealized Self SErving Rational Mark: Will it be interesting?
REVERB Mark: You better be interested! It's the Presidential Debate!
PISSER Mark: I don't follow that.
REVERB Mark: You're a functioning, adult member of the society now. You live in a democracy, and that's a great privilege. It's your responsibility to take part in the governance of this nation, and you didn't even vote in the last election! Now you better turn on that debate.
PISSER Mark: Given my demographic, occupation, community, and personal interests it's not difficult to guess which candidate I support. I doubt that watching the debate will change that.
REVERB Mark: That's not the point. Even if you know who you're going to vote for, you still need to stay well informed.
PISSER Mark: Why?
REVERB Mark: Because you're more privileged than any but the tiniest fraction of human beings alive, and yet you take it all for granted. You live in a nation where you personally can have real, meaningful participation in the political process, and it's your responsibility to exercise that right.
PISSER Mark: I admit I frequently take the perks of life in America for granted, but I don't see why following politics is a responsibility, rather than a personal choice. I'd rather do something else.
REVERB Mark: What do you think politics is, then?
PISSER Mark: A hobby.
REVERB Mark: A what?
PISSER Mark: A hobby. A pastime. Like being a sports fan. You can read about sports in the newspaper, listen to experts talk about it on the radio. Then you can choose which team you like, and "support" them by buying bumper stickers and paying for admission to their games. All the same applies with politics, except the bumper sticker becomes a sign on your front lawn and the ticket to the game becomes a vote. They're both just ways for people to have something to get riled up about without doing much harm.
REVERB Mark: Not an apt analogy at all. Sports are just games. It doesn't really matter who wins the World Series. But it sure matters who wins the Presidency!
PISSER Mark: A little bit, maybe. My job prospects might be a little dimmer with one candidate in charge than the other. Their differences on social issues could affect how pissed off the people around me are. The foreign policy they enact will cause me to read different headlines in the news.
REVERB Mark: That's not just news items. All that stuff is really happening out there, to real people. It's not just all about you. To some people it makes a huge difference who's President.
PISSER Mark: But we're discussing my personal decision on whether to watch the debate.
REVERB Mark: Look, I know what you're thinking. You're thinking, "my vote is just one gumball in Candy Land. Doesn't make a bit of difference. The vote in my state always goes to the same party, and one more vote cast either way can't change the result." But you know what? One hundred thousand votes can change the result. And what if one hundred thousand people just like you are using the same logic?
PISSER Mark: Are you saying that if I go to vote it will change one hundred thousand lazy people's minds?
REVERB Mark: You know that's not what I'm saying. I'm saying you shouldn't vote because you think your particular vote is going to make all the difference. You should vote because it's a statement that you give a rat's ass about the future of this country! Not just for yourself, but for everyone you share this country with. And since the US is a leading world superpower, everyone you share this Earth with. And before you vote, you need to be informed. So you need to watch the debate.
PISSER Mark: Watching just one debate won't make me informed. Becoming a competent participant in democracy is a significant commitment.
REVERB Mark: One that every adult American ought to make.
PISSER Mark: Have you met many adult Americans? On the whole, they're not the ones I want making any important decisions. Basically, ability to choose a good Presidential candidate will undoubtedly fall on a curve like this:
So let the people who are innately interested in the whole process, and can do a good job of it, pour their efforts into politics.
REVERB Mark: All the more reason for you personally to get involved. You're way at the top of that curve. Remember high school? You don't want to common man making the decisions - then YOU better get involved and do something about it.
PISSER Mark: That's not really my point. I don't want to invest huge amounts of time and energy into becoming an "informed citizen", which by the way makes the dubious presupposition that it's possible to make a good choice from behind a computer screen, because it doesn't interest me much. Even if I did watch the debate, I don't know enough about foreign policy or economics to decide who I agree with. I'd inevitably fall back on judging demeanor and rhetoric. I could do more good by putting that effort into something I'll enjoy.
REVERB Mark: Like what? What can you do that's more important than your civic duty?
PISSER Mark: I don't know... plant a tree or something, I guess.
REVERB Mark: Oh come on. You've never planted a tree in your life.
PISSER Mark: I did, actually. I planted the free pine tree seedling they gave me third grade on Arbor Day. I think dad accidentally ran over it with the lawn mower.
REVERB Mark: Not the point. That's not what you will do with that extra time. What you will do is read stupid websites or play flash games, or something inane like that.
PISSER Mark: Or maybe I'll study for a while. Or work on that project I had planned, or...
REVERB Mark: Oh please. You've already got three tabs open and they're Pacman, SomethingAwful.com, and then something more awful than that.
PISSER Mark: sigh. Where is it streaming from?
Friday, September 26, 2008
What does a determinant have to do with a cross product? pt. 1
You're probably aware of a nifty mnemonic for calculating a cross product.
If you take a dot product of the above expressions with another vector, you'll get
What a crazy magic coincidence that vector products and determinants can be calculated the same way!
I won't say magic coincidences don't exist, but I will say that when you find one it may behoove you to look a little deeper. There is a deeper connection.
I'll hint at it today by posing a certain problem, then showing that the problem has a solution whenever either expression, the determinant or the vector product, is nonzero. So is it two different conditions that show the problem can be solved, or the same condition in different guises? We'll deal with that later.
Imagine you are given four random vectors in space,
. Then you are asked to express the fourth as a linear combination of the first three. That is, find 
such that
I won't solve the problem. I'm just asking for a test to see if it's soluble (besides throwing water on it). Especially, I want to see if it's soluble for all possible, not just special cases.
which is a system of linear equations that can be written in matrix form as
(Aside: If you know how to tell to make the vertical columns the same size in those matrices, drop me a comment.)
If you've taken linear algebra, you'll recognize that the problem can be solved only if the above matrix has a non-zero determinant. That same determinant is the one we were interested in earlier, since it's also the value of
and define a plane. They're two independent vectors (I'm assuming they're independent - if they're dependent, you might as well just get rid of one of them for the purposes of solving this problem, but it's clear that in general that won't work) in a two-dimensional space, so they form a basis for that space.
Break down into two parts, the part in the plane of and 
, which we'll call 
and the part perpendicular to the plane, .
and are irrelevant to the perpendicular part of the equation, so we're left with
We need to find
, so we better dot with a unit vector in the perpendicular direction.
Combining the previous two equations lets us solve for.
Unless is by extraordinary chance in the plane of and (and we're interested in the general case, not special ones), we need to be finite, meaning needs to be nonzero. This condition also has built into it the requirement that and be independent, so it's a sufficient condition for the problem to have a solution.
The equation
can be solved for any whatsoever if
also, the same is true if
That doesn't prove the determinant and the vector product are identical, but it's hinting at why it's true. Tomorrow I'll probably post a little about what determinants are, and the geometrical interpretation of the vector product.
If you take a dot product of the above expressions with another vector
What a crazy magic coincidence that vector products and determinants can be calculated the same way!
I won't say magic coincidences don't exist, but I will say that when you find one it may behoove you to look a little deeper. There is a deeper connection.
I'll hint at it today by posing a certain problem, then showing that the problem has a solution whenever either expression, the determinant or the vector product, is nonzero. So is it two different conditions that show the problem can be solved, or the same condition in different guises? We'll deal with that later.
Imagine you are given four random vectors in space,
I won't solve the problem. I'm just asking for a test to see if it's soluble (besides throwing water on it). Especially, I want to see if it's soluble for all possible
Using A Basis
Set up a standard Cartesian basis in three dimensions, and write out the vector equation in componentswhich is a system of linear equations that can be written in matrix form as
(Aside: If you know how to tell
If you've taken linear algebra, you'll recognize that the problem can be solved only if the above matrix has a non-zero determinant. That same determinant is the one we were interested in earlier, since it's also the value of
Without a Basis
VectorsBreak
We need to find
Combining the previous two equations lets us solve for
Unless
Summary for today
The equation
can be solved for any
also, the same is true if
That doesn't prove the determinant and the vector product are identical, but it's hinting at why it's true. Tomorrow I'll probably post a little about what determinants are, and the geometrical interpretation of the vector product.
Monday, September 22, 2008
A X (B X C)
Any intro to physics course will teach you the identity
It's easy to prove by explicitly writing out each vector in component form, cranking through both sides of the equation, finally getting the same thing on each. But there's some insight to be gained in looking at the identity from a purely geometric point of view. Warning: the following is a bit dense, but also short. Go one sentence at a time and you should be fine.
Together, and are a basis for a plane. is perpendicular to the plane. Take that perpendicular vector product, cross it with , and you get a vector perpendicular to the perpendicular to the plane. That's back in the plane again. So we know that the double cross product is in the plane somewhere. But any vector in the plane can be written as a sum of and (that's what it means to be a basis), so we only need to find the right coefficients. Those coefficients come from requiring that we find a combination of and perpendicular to . Just dot the expression 
with, and you'll get zero.
That proves both sides of the equation
point in the same direction, but doesn't prove they have the same length. It's not crazy difficult to prove the they have the same length without using a coordinate system, but I don't have a neat insightful trick for it, either.
I started by noting that you could break down into perpendicular and parallel components to the plane defined by the other two vectors, and that as far as the identity is concerned the perpendicular part of doesn't matter (plug it in and you'll see). Then, because both sides of the equation are proportional to the lengths of all three vectors, you might as well divide by their lengths to make them unit vectors. All that's left is to find the lengths of both sides of the equation in terms of trig functions of the angles between the vectors. You have tree angles between the coplanar vectors. Write one as a sum of the other two, expand out all the trig functions, and fiddle around with it a bit until you see that it's actually a trig identity. That gets the final part of the theorem in there.
It's easy to prove by explicitly writing out each vector in component form, cranking through both sides of the equation, finally getting the same thing on each. But there's some insight to be gained in looking at the identity from a purely geometric point of view. Warning: the following is a bit dense, but also short. Go one sentence at a time and you should be fine.
Together,
with
That proves both sides of the equation
point in the same direction, but doesn't prove they have the same length. It's not crazy difficult to prove the they have the same length without using a coordinate system, but I don't have a neat insightful trick for it, either.
I started by noting that you could break
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