Recently, I was talking about vectors. At that time, I had to stop and recall how I had been representing vectors. Ideally, I should stick with the same notation I used in Basics: Vectors and Vector Addition. But let me go over the different ways you could represent a vector.
Maybe this is too obvious, but it had to be said. You can represent vectors by drawing them. In fact, this is very useful conceptually - but maybe not too useful for calculations. When a vector is represented graphically, its magnitude is represented by the length of an arrow and its direction is represented by the direction of the arrow. Here is an example:
I think the biggest negative to this representation (other than being difficult to get numerical answers for adding) is that it is not too easy to represent in 3-dimensions. For the following representations, I will try to relate them to the graphical representation.
Magnitude and Direction
In algebra-based courses, maybe this format is popular. Basically, you just give the magnitude of the vector and the angle (from the positive x-axis) that the vector is pointing. Here is an example (using the same vector from before):
And in magnitude-direction format, it would be:
I am not too found of this format. First, if you want to add vectors, you need to find components. Second, students often get confused with this angle always being measured from the same axis (it doesn't have to be the x-axis, that is just what is common). Oh, if you want to do this for a 3-D vector, it really isn't worth it. You would need two angles. Well, in some cases it might be worth it.
With the component method, the idea is to just give the amount the vector is in each of the coordinate directions. Here is an example.
Hold on. I am not finished. Yes, I wrote these components as vectors so that:
Often you will see textbooks sort of stop here. In this case they may say something like:
It is important to realize that this notation is NOT the magnitude of the vector Fx and Fy. The magnitude of a vector has to be a positive number. To really use these, you need unit vectors. This is what they look like:
The little ^ over the x means that it is a unit vector. A unit vector is a vector that has a magnitude of 1 with no units. This means that the Fx vector could be written as:
And maybe now you can see why that negative sign is important. The vector Fx is in the opposite direction as the x-hat vector and that is why you need a negative sign. So, using this notation, you could write the vector F as:
Some textbooks like the you i, and j instead of x and y - this would look like:
Same thing, different looks. Don't forget units though. Vectors have units, if you leave them off you are probably a mathematician (just kidding). Also, this notation can be expanded to three dimensions by adding a z-hat or k-hat component. Another nice thing is that these vectors are all set up and ready to add. If you have a vector in component notation you are ready to rock.
I guess the reason textbooks use the magnitude-direction format some is that it may be easier to relate to real life. In real life, I would measure the magnitude and direction of a force and then have to calculate the components.
Same thing, but another way
I really like the physics textbook Matter and Interactions by Ruth Chabay and Bruce Sherwood. The way that textbook consistently represents vectors is as:
I like this notation. It is short and it emphasizes the components as well as the idea that all forces are 3-dimensional. The short thing is really good for lazy people like me. Also, it matches up really nicely with vectors in vpython. Here is how I would write that vector in vpython:
Vectors have units, if you leave them off you are probably a mathematician (just kidding).
But, yeah, it's kind of true. Mathematicians tend to worry about units somewhat less than physicists (even though a few of my colleagues are adamant about them). For example, when we're computing definite integrals to find the area under a curve, I find it unnecessary to write "units squared" as many of my students feel compelled to do (they must have learned it in their physics classes -- "Do this when your math teacher fails to specify the units!"). Not sure.
Sorry for that :P But a vector isn't necessarily an "arrow", but may also be something abstract like a function ;)
Physics 101 continues to be one of my favorite courses ever. I'd point out that you can represent forces in N dimensions with components. For 3 dimensions, k is often a favorite, as in
ââ = (-9.9)Ã®+(9.9)Äµ+(0.0)Ä¸
(I couldn't find a nice F or k to match the symbols, but you get the idea).
I have one question...If somebody says that vector A is (6i-9j) then does that geometrically, in x-y plane, means that vector has the coordinate of (x,-y)= (6,-9). And also does it mean that magnitude of x-component of vector A i.e Ax = 6 and y-component Ay=-9.....I was facing difficulty in interchanging a vector equation in normal equation and vice versa
if Ax implies a dot product, then yes, you've got it right. The i and j notation is the same as the x and y notation in most conventions. The notation for z is 'k'. Just remember the right hand rule (curling your right hand's fingers from positive x(i) to positive y(j) in your coordinate system should make your thumb point in the positive z(k) direction) and you'll be fine. It's like anything else, it just takes practice!
Not to add any useful content, but for those who haven't seen it, this is probably the funniest use of Bra-ket notation I've ever seen
Ruth Chabay and I would be happy to take credit for the notation used in our physics textbook but in fact it was our students when we were at Carnegie Mellon who told us that their calculus textbook used this notation. We adopted it because it's easier to write (as you say) and it has the flavor of indicating that a vector is a single thing, not three different things. Whether it is this simpler notation, or the use of vectors in the VPython 3D programming environment our students use for computational modeling, or both, we do observe that our students develop a more sophisticated notion of the concept "vector" than we had seen before. For example, before using and before VPython, even strong CMU honors students had a hard time using unit vectors even in the 2nd semester (E&M) course. Now even not very well prepared average students at NCSU are quite competent with unit vectors a few weeks into the first semester (mechanics) course. We put some emphasis on the fact that any vector can be factored into the product of its magnitude times its unit vector, which encourages a "divide and conquer" in such tasks as calculating a gravitational force in 3D between stars: first calculate the relative vector r = r1-r2 from star 2 to star 1, calculate |r| and use it to calculate |F|, calculate rhat = r/|r|, then assemble the vector -|F|rhat.
Incidentally, showing the arrow representation works fine in 3D in the VPython context; it's just hard to draw by hand on paper. Also, while F = vector(-.9.9, 9.9, 0) is in VPython an abstract mathematical statement, the statement arrow(pos=vector(3,4,-2), axis=F) creates an arrow object whose origin is at (3,4,-2) and whose axis has the length of the magnitude of F and direction of the direction of F. Here is an opportunity to distinguish operationally between an abstract vector and its representation as an arrow. It even brings up the issue that in order to fit on a screen whose width and height might represent 2 m by 2 m, an arrow representing a force of 500 N must have axis=(2/500)*F; normally textbook don't point out that in diagrams arrows representing force or velocity or momentum etc. were scaled to fit onto the spatial diagram.
Finally I'll point out that in 3D sines and cosines don't get you very far but direction cosines are great. It's easy to show that the components of a unit vector are the cosines of the angles from the vector to the +x, +y, and +z axes. It seems to us that even in two dimensions it makes sense to emphasize unit vectors and direction cosines, to remove the endless confusion students have about whether to use a sine or a cosine: just use the simple rule of always using a cosine to the +axis.
For more info, see matterandinteractions.org.
Oops! I should have previewed. Where I wrote "bra x,y,z, ket" this fails to show up because it was treated as html coding. In particular, the following was intended: "For example, before using bra x,y,z ket and before VPython...."
I would like to make a plug for the graphical representation. Now, I teach high school, so maybe you are expecting your students to come into your class already possessing the kinds of skills I am teaching. So, I'll qualify my comment by noting that I am talking about teaching vectors to 15-17 year-olds. I used to teach sophomores the graphical method of adding and subtracting vectors (using protractors and a scale on the page) and then quickly move on to breaking vectors into components. The students tended to immediately drop the graphical method for the component method, but not for the reasons you state. They did it because with the component method they had an algorithm... they didn't have to think about the vector as a physical quantity anymore... they had an "automatic" technique (and who doesn't have more important things to do with their brain at 15?). Starting almost immediately, the students became more and more disconnected from the physicality of a vector. David Hestenes calls this "componentitis." So now I delay teaching component form to sophomores until they have pushed arrows representing vectors around on the page for a good three months. The results are remarkable. Even when they learn about adding and subtracting in component form (and quite a few discover this on their own, which is priceless), they are much more apt to draw vectors on the page, visualize what the vector algebra will likely produce and check their component form answer against their physical intuition. Some industrious students solve problems both completely graphically and in component form, just to make sure of their answers (they also tend to be the ones wearing belts AND suspenders).
I am strongly considering introducing the bracket notation to sophomores, too. I think it will help them to focus on the vector as one entity, even after it is "broken into components." Also, it may help them to simply keep track of what they are doing. I use the bracket notation with my second year physics students, anyway (I use Matter and Interactions for this class). My second year students (now that they have received the more "physical" training in vectors) are very adept at thinking geometrically about dot products and cross products. Students thinking this way (even if they are in first year calculus) tend to make mincemeat of their peers in multivariable calculus when it comes to using and understanding cross products and dot products.
I agree completely with Mark on the importance of the geometrical approach to vectors. And no, our college freshmen don't come to us with such skills (nor do we do a particularly good job on this aspect in the limited time we have to devote to this). I didn't say anything about the geometrical aspect only because I was focused on the notation issue that Rhett had emphasized.
A related issue is that we find it difficult to get students to draw anything, let alone arrows representing vectors. We physicists can hardly do any reasoning at all without diagrams, but students seem to think that diagrams are unnecessary decorations, not tools, and must be drawn only to please the teacher and get the 4 points awarded for the diagram part of the test question.
That said, we have some anecdotal evidence that a problem is that students are not taught how to draw and use diagrams. There are a lot of hidden assumptions in the way we draw diagrams. For instance, there are situations where it is very important that a square be shown as a square, not a rectangle, and other situations where it's irrelevant whether the thing is represented by a square or rectangle, and knowing which situation it is may require a lot more knowledge than the student has, so the student feels a bit paralyzed, not knowing the rules for drawing a useful diagram.
While it has been some time since you wrote these comments, I would like to say that the notation for a vector, , is the notation found in Maple and the one I have adopted. For example: F:=<3.2, 2.5, 1.7>;
I have also used square brackets as a delimiter for units. For example, to say a table is 1.7 meters long, I write: L = 3[m]. This avoids a confusion of interpreting the length as 3*m where m is some variable. Interestingly, Maple uses very similar notation for units, though closer to double square brackets: [[m]].
Excellent. Your tutorial really helps to boil down the complexities of vectors and their notations to form anyone can easily understand. Thanks, a million.