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/