How do I define a 3D line that does not leave the horizontal plane?



Answered question


If I want to define the line y=2x when discussing 3D lines this is actually a plane. We should be able to turn it into a line with cartesian equation x=2y=z/0 but that would give us something undefined. How do you resolve that problem?

Answer & Explanation

Xavier Jennings

Xavier Jennings

Beginner2022-09-04Added 9 answers

The general rule-of-thumb is that in n-dimensional space R n , every equation you add cuts the dimension by 1
In the plane, y=x gives a line. A system of 2 equations like x=y and x=3 determines a point.
In R 3 , y=x gives a plane. The system x=y and y=z defines a line, and something like x=y, y=z, and z=3 gives just a point.
In 7-dimensional space, y=x gives a 6-dimensional hyperplane, you need 6 equations to describe a line, etc.
All you need to do here is write x = 2 y and z=0. I suppose with some elbow grease you could force this into one equation like ( x 2 y ) 2 + z 2 = 0, but I'm not sure this is an improvement.


Beginner2022-09-05Added 6 answers

Typically you would describe it parametrically:
( x , y , z ) = ( t , 2 t , 0 ) , t R
To elaborate on this, two pieces of information that describe a line are a starting point (which you can choose anywhere on the line) as well as a direction vector (which you can scale by any non-zero number):
( x , y , z ) = ( p x , p y , p z ) + ( v x t , v y t , v z t ) , t R
Since this notation starts to get cumbersome, we can simplify it by writing
x = p + v t , t R
where x represents (x,y,z) and similarly for p and v.

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