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?

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Xavier Jennings · Accepted Answer

The general rule-of-thumb is that in n-dimensional space             R        n  , every equation you add cuts the dimension by 1In 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&#039;m not sure this is an improvement.

KesseTher12 · Accepted Answer

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.

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

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