The Power of Dimensions

27 January 2015

357 words

I suspect the majority of people who have any use for this already know about it and use it daily, but I have only recently realized how cool this is.

You know how you always forget what the relationship between c, λ, and ν is? Like, they give you the energy of a photon and you’re supposed to calculate the wavelength? You know E = h ν, but now you have to figure out how to convert the frequency to a wavelength. Now, instead of looking the formula up on Wikipedia, how about using dimensional analysis? You write down the unit of λ, which is m, and then you only need to figure out how to construct that unit with a speed and a frequency. Speed is m s^–1 and frequency is s^–1, so to cancel the seconds out, you divide the speed by the frequency and get m s^–1 s = m, from which you can easily see that λ must be c / ν. And you’re done.

Another good example is when you have an exponential function or a sine or cosine. You know the argument for these functions must be dimensionless, so if you’re not sure what factors you have to put into the argument, you can just keep throwing stuff in there until all the units cancel and it’ll probably be right.

Or if you want to sanity check your calculations you need only look at your dimensions. When you see a sum of, say, a length and an area, you know you’ve done something wrong.

The same applies of course to the other kind of dimension — the one in vectors. When you try to put a three dimensional vector in an exponential function, you have a problem. You’re probably missing another vector to form a dot product with. Or if you’re trying to add a vector and a scalar, again, you know you’re missing something.

It seems pretty simple, but I was surprised how useful this is once you get the hang of it.