Question

Similar to the way "y=mx + b" provides a general form of a line in two variables, is there a general form for a line in three variables? I am OK remembering planes in three variables, but I'm getting mixed up thinking about lines in three variables.

Thanks!

Answer 1

Here you have to make a distinction between the number of variables of the object and the number of dimensions of the space in which the object is embedded.
So you could have y = 2x -3 is a line in 2D but what is it in 3D?
Or a circle x^2 +y^2 = 4 is a circle in 2D (R^2) but in 3D?

In 3D, as with 2D, there are a number of different ways to describe a line, vector, parametric, symmetric.

I wouldn't call any of them a general or standard form (I don't think that any for should be described as general or standard in 2D either, to be honest).

It just depends on what you are trying to do...

Answer 2

Just trying to re-learn a bit... :) so, if you had a 3D space like x-y-z, and you had the 2D line y = 2x - 3 in the x-y plane, do you have to do anything to "talk about it" (please excuse loose terminology) when you are setting up systems of equations with 3 variables?

Like intersection of that line with some other plane, etc...

Answer 3

the line itself doesn't depend on z, so I get why there is no z term in it.

But in general, a line passing through 3D space could vary in all three dimensions, so it would need x, y, and z terms, I would think...

Answer 4

As long as you say that the z cordinate is 0 (or as you said, in a specific plane), no problem.

f u don't say, then, it will be a plane!

Answer 5

ah... the line projected in 3D becomes a plane... got it. so you need to state a value or range or whatever for the 3rd dimension to make it other than the infinite plane that includes the original line

Answer 6

symmetric equation in 3D (x-x_0)/a = (y-y_0)/b = (z-z_0)/3

parametric (common) x = x_0 +ta, y = y_0+tb, z= z_0 +tc

Answer 7

"ah... the line projected in 3D becomes a plane... got it."

Yes (and your circle would be a cylinder)

Answer 8

sure... I'm fine with visualizing it.. I was trying to build intuition on lines in space, similar to how you can get a sense of lines and curves in 2D when you have a "deep" understanding of slope, derivatives, inflection, etc.

Answer 9

Lots of things generalize in an intuitive way, particularly the gradient (function).

The best way to gain intuition is to play about with (2D) surfaces in R3.

Answer 10

Looking at systems of 3 variables, I don't have as deep an instinct for what the spatial representation is... I could plot it, but just like you hate for algebra 1 students to be stuck forever just plotting ordered pairs instead of "seeing" the lines, I was brushing rust off my 3D skills

Answer 11

Thanks for the help...

Answer 12

ur welcome