asher63 (asher63) wrote,

Thinking About Numbers

To keep my mathematics circuits free of cobwebs I've been reviewing facts from algebra. Looking at the Law of Cosines, I noticed for the first time that it contains an expression familiar from vector math. Under the Wiki entry for 'Dot Product' I found this very cool proof: if you let vector C represent the vector between the endpoints of A and B - that is, A-B=C vector-wise - and you square both sides (using the dot product), you get the Law of Cosines. That is, C dot C equals (A-B) dot itself; and you expand the latter as a binomial square-of-difference. That's where the "-2AB cos c" term comes from, it's the same as "2 * (A dot B)".

I've started poking through my old Dynamics textook too, to see if there was anything forbiddingly difficult therein, and I don't think there is. Just for kicks I skipped ahead to take a look at Sample Problem 17.5, where you've got two rigid rods hinged together with the end of one hinged to a surface and the free end on a frictionless roller. The strategy involves finding something called the "instantaneous center of rotation" which is given by drawing lines perpendicular to the moving points (i.e. along the radii of an imaginary wheel) and finding where they intersect. It's a cool concept, and I will probably think about it every time I see a book slide down flat after losing its grip on the bookend. Anyway, if I ever manage to earn a college degree before I die, I'm still interested in engineering.

Tags: geekery

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