Why OH Why_3D Parametric Equations

[karen03grae]

Our lecture today covered Equations of Lines and Planes in 3D.

Is this the only approach to learning line and plane equations in 3-d?

Honestly do we need r = ro + t*v?

To me this seems like a very hard way to learn equations of lines and planes.

Maybe I should learn it to be a more well-rounded Cal. student.

Any suggestions on if this is the only way to determine Eq.s of lines/ planes?

Thanx!

 

[arildno]

You are approaching this with the wrong attitude.
Rather than bemoaning your initial problems with this, you should ask yourself:
What are the advantages of thinking of lines in terms of parametrizations?
The immediate advantage, is that essentially all types of CURVES (not just straight lines!) is a ONE-DIMENSIONAL object!
This one-dimensionality is expressed in that a single parametrization variable is needed to describe the curve.

Further, a surface/area is essentially a two-dimensional object; you can describe any surface/area with two independent parametrization variable.

So, you have through parametrization an elegant unification of many seemingly diverse objects.

And yes, this formalism is absolutely needed.

 

[matt grime]

Or, one may say "I learned a definition today. I didn't like it", well, sorry, it's a definition. It's not the only one, but it is one the others are less intuitive. Deal with it. Feel free to offer another one: you imply you know other equations of straight lines in R^3. [crap analogy: I learned that chien was dog in french today. why? dog is such a shorter word, and french is a romance language so why isn't it closer to canus?...]

Incidentally, do you get any geometric intuition from:

(x-a) wedge b=c, or (x-a)/p=(y-b)/q=(z-c)/r

or understan that these are just ways of DESCRIBING geometric shapes, not defining them.

Show that two arbitrary lines do or do not intersect, eg the line L(1) passing through the points (1,1,0) and (0,0,2) and the line L(2) passing through the points (1,0,0) and (3,2,1)...

 

[karen03grae]

It's not the only one, but it is one the others are less intuitive.

Okay, I was wondering if there was a more intuitive way out there. But I suppose this way is fine.

Incidentally, do you get any geometric intuition from:

(x-a) wedge b=c, or (x-a)/p=(y-b)/q=(z-c)/r

No.

K, I'm going to work on my homework now.