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: