We have a big problem-solving poster on the MLC wall that gives students advice for solving problems. One of those pieces of advice is that to decide what to do for your current problem, you could look at other problems for inspiration. Yesterday I saw the dangerous results of what happens if you look at other problems for instructions rather than inspiration.
Across the day I talked to several students about Statics, which is an engineering physics course concerning situations where objects are not moving (ie “static”). At least three of them had put their answers into the computer-marking system and found that they were wrong, and so they wanted help to figure out why. As always, I asked them to tell me more about the problem and how they solved it.
The students showed me the problem they were trying to solve, and then pulled out their lecture notes and showed me how they found a problem that was similar and followed the same procedure. Normally, this would make me extremely happy – students actually looking at their actual lecture notes independently? Bliss! But unfortunately, what they had done was notice how the example had the same letters in it as their problem and put the values of those letters from their problem in the formulas they saw in the example.
The big problem was that the forces in the book’s examples were perfectly horizontal, but the ones in their assignment were at an angle, so they couldn’t just put things in the formula. Oh dear.
These students had interpreted the example as a list of instructions for precisely what to do if they saw a similar picture with similar letters in it. They had interpreted the process of solving Statics problems as finding the right formula and putting things in the right places. Instead, Statics problems are more about making your own formula from the structure of the problem situation itself. When you read a Statics example, you are looking for inspiration for how to think about the problem, rather than for instructions for how to do it.
I am very glad now that when we chose the words for our poster, we used the word “inspiration”. In the future I will try to highlight that aspect of it a bit more, so that students can be looking for the right things when they perform the good practice of looking for examples in the notes.
This comment was left on the original blog post:
Terry Bennett 10 April 2015:
Great post/observation David,
I’ve come across the same thing in class – “.. are you trying to reuse the equations on pg44 of my notes? Did you draw a free body diagram of this particular problem?”.
Next step: the reflection “what can I do better next year to avoid/reduce this misconception?” (answers on a postcard please …). Thanks for the help in identifying this “recipe trap” I’ve inadvertently introduced.
I had a long chat with one of the students the other day about rotation matrices. They had come up in the Engineering Physics course called Dynamics as a way of finding the components of vectors relative to rotated axes. He had some notes scrawled on a piece of paper from one of my MLC tutors, which regrettably were not actually correct for his situation. I know precisely why this happened: rotation matrices are used in both Dynamics and Maths 1B, but they are used in different ways (in fact, there are two different uses just within Maths 1B!). It’s high time I made an attempt to clear up this confusion, especially since three more students have asked me about this very issue in the last week!
In Maths 1B, you learn about Linear Transformations, which are a special kind of function that you enact upon vectors in some dimension to produce vectors in some dimension. It turns out that all linear transformations can be described by representing your vector as a column of coordinates and multiplying it by a matrix. Each linear transformation has its own matrix that works for all the vectors it acts upon. Rotations happen to be a type of linear transformation and in two dimensions there is a formula based on the angle you rotate that tells you what the matrix is. I’ve included just such a matrix in the picture here.
One reason this works is because multiplying a matrix by your standard basis vectors of (1,0)T and (0,1)T gives you the first and second columns of your matrix respectively. But multiplying by the matrix has the same effect as the rotation transformation, so to figure out what these columns actually are, all we have to do is rotate the points (1,0) and (0,1). If you do this, then because of trigonometry, you get the two points (cos θ, sin θ) and (-sin θ, cos θ), which are indeed the columns of the matrix.
Let’s just make sure we know what’s going on here before we move on: You have a point in the 2D plane, you take its coordinates as a column, you multiply this column by the matrix, and you produce a new set of coordinates, which is a new point. So your matrix in effect moves your point from one place to another. The point with coordinates (1,0) moves to the point with coordinates (cos θ, sin θ); the point with coordinates (0,1) moves to the point with coordinates (-sin θ, cos θ).
So now we have that a rotation matrix has cos θ on the main diagonal, sin θ in the bottom left corner and -sin θ in the top right corner. And it tells you where a point moves to under a rotation of θ anticlockwise. (It’s worth noting that it also works perfectly well on the components of vectors imagined as arrows.)
The problem is that over in Dynamics, a rotation matrix does not look quite like this! In particular, the minus sign is in the opposite corner. Why?
The answer is that in Dynamics the rotation matrix is not a description of a transformation of the points or arrows themselves, but a description of how their coordinates change when you transform the coordinate axes. The points themselves don’t move at all, it’s the coordinate axes that move and we just relabel the points with new coordinates.
The reason this works is again because of the standard basis vectors. The point (1,0) has its coordinates recalculated according to the new axes, and its coordinates turn out to be (cos θ, -sin θ); while the point (0,1) also has its coordinates recalculated and its coordinates turn out to be (sin θ, cos θ).
You may notice that this is precisely what the coordinates would have been if you did rotate the points themselves, but in the opposite direction to the original rotation matrix. This makes sense. If you turn your head to match the new coordinate axes, then this is precisely what has happened. Basically, if you rotate the coordinate axes one way, the points “move” the other way relative to the axes.
And this would be the end of the story, except that in Maths 1B you also rotate coordinate axes, and yet the rotation matrix is somehow still not the same as the one in Dynamics! Why?
The reason is that in Maths 1B we rotate axes in the context of equations of curves, and this is quite a different situation from when you rotate axes in the context of the points themselves.
Imagine I have an equation which describes a curve. A point is part of the curve if its coordinates satisfy the equation, and it’s not part of the curve if its coordinates don’t satisfy the equation. But what if I relabel all the points with new coordinates according to a new set of axes? I want an equation for my curve so that a point is on the curve if its new coordinates satisfy the new equation. How do I achieve that? Well I do already have an equation, it’s just in terms of the old coordinates. So if I have a point in the new coordinates, to tell if it’s in the curve, I just need to figure out what the old coordinates are and sub them into the old equation. It ought to be possible to make one equation that encompasses both of these actions – the transferring to the old coordinate system and the subbing into the old equation.
Did you notice what happened there? In order to create an equation that described the same curve relative to the new axes, I had to begin with the new coordinates and transform them into the old coordinates. Let me repeat: I had to go from new to old. The coordinate transformation matrix in Dynamics goes from old to new. To go in the opposite direction I have to have the minus in the opposite corner.
So that’s why the matrices are different. In Dynamics you are moving the axes but not the points, and finding new coordinates for the points. In Maths 1B you are moving the points, not the axes, so the rotation appears to be in the other direction. Or alternatively in Maths 1B you are moving the axes, but you already know the new coordinates and you want the old ones, so you actually are doing the calculation in the opposite direction.
Students in Maths 1M are learning the cross product at the moment. This is a way to multiply two vectors in 3D space – let’s call them v1 and v2 – to produce a new vector, which is called v1 × v2. The length of this new vector is related to the lengths of the two original vectors and the angle between them, and the direction is perpendicular to both of the original vectors. However there are two possible directions it could point and still be perpendicular to both. We need a consistent way to choose which of the two options to use, and this is provided by the so-called “right-hand rule”.
I was taught the right-hand rule as per the picture here: you extend your thumb and forefinger as far as they go, and then stick out your middle finger. Then arrange it so your index finger points in the direction of v1 and your middle finger points in the direction of v2 . Then your thumb must point in the direction of v1 × v2. I have very strong memories of looking around in my first-year Physics exam to see people in various contortions as they used the rule.
It’s a cute little rule and does the job well, but in fact it is not the only correct version of the right-hand rule. Often it is taught as if it’s the only possible way to do it, but to be honest it’s just a mnemonic so some other version is actually ok! Especially if you happen to find the other version easier to use and remember. I thought I’d put the various versions I know here to compare the alternatives.
First, let’s look at a very slightly modified version of the original three-finger version. I don’t know about you, but it tends to hurt my hand to have the middle finger out but the other two curled under (it reminds me of the pain of trying to do the Vulcan salute too often). So I like to do the version shown on the right, where I point all three of the other fingers outwards. It’s not really different from the first version, but it sure is less painful!
The next version is pictured here on the left. The fingers are in the same arrangement, but different fingers relate to different vectors. In this alternative version, the thumb is v1, the index finger is v2, and the other fingers are v1 × v2. Personally I find this one easier to use and more realistic because you can move your thumb to indicate smaller and bigger angles between v1 and v2. Other people like the first version because they think of their index finger as their first finger and so it makes sense to them for it to be the first vector.
From this alternate version, we can ease the strain on our hand just a little more by realising that we don’t technically need the other three fingers to point outwards because our palm always faces that way anyway. So here is the “palm” version of the right-hand rule on the right: your thumb is v1, your fingers are v2, and v1 × v2 points out of your palm. This is my favourite, and not just because it’s the easiest to actually manipulate your fingers into shape! It gives to me a real sense that v1 and v2 are creating a plane and the cross product is pointing out of it.
Even though the palm version is my favourite, there are two more versions I know about, so I’ll mention them too. The first I call the “sweep” version and it’s pictured on the left: you orient your four fingers towards v1, and then you sweep your fingers towards v2. If you do this, then your thumb will point in the direction of v1 × v2. There is something about the dynamic nature of this version that I do like — you really get the feel that the cross product is actually doing something to the two vectors because you’re moving your fingers. It does however take a greater leap of imagination than the others.
The final version I call the “curl” version and it’s shown on the right. It has the same idea as the sweep version, but you basically represent the sweep statically. You imagine v1 and v2 in a plane and you curl your fingers to represent the direction you need to rotate to go from v1 to v2. Then your thumb will point in the direction of v1 × v2 . This one requires a lot of imagination again, but it does have an advantage that it is the same arrangment of your fingers that you use in Physics to find the direction of a torque or the direction of a magnetic field based on a flowing current. Interestingly, this is one of the most common to find by searching on the internet, and is the one described on Wolfram MathWorld.
So there you go: that’s six versions of “the” right hand rule. Whichever of them makes most sense to you and causes least pain is fine for you to use – you can even make up your own if you like! But do remember other people may use different ones so take care when communicating with others to tell them how your version works!
