I'm back, and I think I did worse than I had anticipated. I may reach the goal required for higher ed in the directions that need significant above-average grades in mathematics, but I'll be lucky if I get beyond that.
In short, I think I lack the kind of observation one gets from long practice. Quite a number of the things that really got me (so I had to work a long time to find out) were algebraic manipulation things like:
Show, using sin(u-v) and cos(u-v) identities, that tan(u-v) = (tan(u) - tan(v)) / (1 + tan(u) tan(v))
The trick, which I found out after too many precious minutes, is to see that tan x = sin(x)/cos(x), then you get
then divide both numerator and denominator by cos(u)cos(v) to get the correct tangents. If I'd been a little more familiar with algebraic manipulation, I would have seen the 1 in the result as a giveaway, a huge sign saying "Here! Here! Something cancels out here!".
As it went, I got seriously pressed for time and I did at least the latter half of the last exercise (which involved fitting a sphere between two planes using vector math) wrong. I was probably supposed to find a point on one of the planes, then project it onto the other to get a 90' line between the two planes, then find the midpoint as the center of the sphere and infer the radius from there. However, instead I picked two points already specified to be on the respective planes and calculated the line between them, then fit the radius so the shell of the sphere incorporated one of the points. This is wrong because if the line is slanted, the sphere will either be too large or too small.
(If I'm lucky, they'll say "okay, you got the line wrong but the way you used it after that was right so we'll only dock points once".)
As it is, I told my parents that if I do end up getting the top grade, I'm going to memorize and sing the entirety of "I Am the Very Model of a Modern Major-General" to them. :)
The first problem you mentioned is one that almost never appears in the text of a book, and almost always appears in the problem section of a book. So yeah, it was just lack of practice that zapped you. On the plus side, you'll probably remember that particular trick forever. :P
The second one mentioned was a linear algebra problem, not a geometry problem. Which is to say, if you had been fast at doing matrix manipulations by hand (like I am, due to excessive practice), and you had known the requisite tricks (and linear algebra is crammed to the gills with tricks), you could have finished the problem quite quickly. In this case, for me the quickest method would have been: (i) find the equation of the plane halfway between the two planes, using an LA trick; (ii) find the equation of the line passing through the center of the sphere and perpendicular to the planes, using yet another LA trick; (iii) using LA trick number three, find the length of the line segment between the two planes, which will be the diameter of the sphere. If you were even more experienced with linear algebra than I am, you could do all three steps in a single calculation, by hand. Knowing you, I bet you didn't even try to use matrix manipulation, though. Too inelegant.
I've taken plenty of tests like this. You did well. Top grade well? I have no idea. But you did well.
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branchandroot
no subject
Date: 2012-06-04 04:47 pm (UTC)In short, I think I lack the kind of observation one gets from long practice. Quite a number of the things that really got me (so I had to work a long time to find out) were algebraic manipulation things like:
Show, using sin(u-v) and cos(u-v) identities, that tan(u-v) = (tan(u) - tan(v)) / (1 + tan(u) tan(v))
The trick, which I found out after too many precious minutes, is to see that tan x = sin(x)/cos(x), then you get
tan(u-v) = (sin(u) cos(v) - sin(v) cos(u)) / (cos(u) cos(v) + sin(u) sin(v))
then divide both numerator and denominator by cos(u)cos(v) to get the correct tangents. If I'd been a little more familiar with algebraic manipulation, I would have seen the 1 in the result as a giveaway, a huge sign saying "Here! Here! Something cancels out here!".
As it went, I got seriously pressed for time and I did at least the latter half of the last exercise (which involved fitting a sphere between two planes using vector math) wrong. I was probably supposed to find a point on one of the planes, then project it onto the other to get a 90' line between the two planes, then find the midpoint as the center of the sphere and infer the radius from there. However, instead I picked two points already specified to be on the respective planes and calculated the line between them, then fit the radius so the shell of the sphere incorporated one of the points. This is wrong because if the line is slanted, the sphere will either be too large or too small.
(If I'm lucky, they'll say "okay, you got the line wrong but the way you used it after that was right so we'll only dock points once".)
As it is, I told my parents that if I do end up getting the top grade, I'm going to memorize and sing the entirety of "I Am the Very Model of a Modern Major-General" to them. :)
no subject
Date: 2012-06-05 03:52 pm (UTC)The second one mentioned was a linear algebra problem, not a geometry problem. Which is to say, if you had been fast at doing matrix manipulations by hand (like I am, due to excessive practice), and you had known the requisite tricks (and linear algebra is crammed to the gills with tricks), you could have finished the problem quite quickly. In this case, for me the quickest method would have been: (i) find the equation of the plane halfway between the two planes, using an LA trick; (ii) find the equation of the line passing through the center of the sphere and perpendicular to the planes, using yet another LA trick; (iii) using LA trick number three, find the length of the line segment between the two planes, which will be the diameter of the sphere. If you were even more experienced with linear algebra than I am, you could do all three steps in a single calculation, by hand. Knowing you, I bet you didn't even try to use matrix manipulation, though. Too inelegant.
I've taken plenty of tests like this. You did well. Top grade well? I have no idea. But you did well.