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: 2012-06-05 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.