Trigonometry in the Third Dimension
Date: 04/30/98 at 07:45:23 From: Brett W. Subject: What does working in 3D change? Hi, I'm trying to figure out an extremely trivial calculus problem that deals in three dimensions. If I want to use trigonometry or calculus in 3D, does it change much? Do I have to use those functions a special way? I've tried just guessing but you can define an angle three different ways in 3D (ratio of x to y, x to z and y to z). Am I on the wrong track? Thanks, Brett W.
Date: 04/30/98 at 08:21:00 From: Doctor Jerry Subject: Re: What does working in 3D change? Hi Brett, Using trigonometry in R^3 doesn't change in principle, but it is somewhat harder to visualize. Mostly, trig enters through vectors. Most of trigonometry is captured in the idea of vector and the dot and cross products. To define an angle, you need more context. The angle between two lines that intersect is not hard to find. If you give the lines in vector form: line 1: r = a+t*b, -oo < t < oo line 2: r = p+s*q, -oo < s < oo where a, b, p, and q are vectors and t and s are parmeters, the angle w between these lines can be calculated from: b.q cos(w) = --------- (|b|*|q|) Note, the period represents the dot-product. The angle between planes is calculated by looking at the angle between vectors normal to the planes. The angle between two curves that meet at a point is calculated by calculating the angle between their tangent vectors. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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