Understanding Line in R^3 Parallel to XY-Plane: Help Needed

[kerrwilk]

What does it mean if a line in R^3 is parallel to the xy-plane but not to any of the axes. I really don't know what this means in terms of how the parametric and symmetric equations of the line should look. Please help. Thanks.

 

[Rasalhague]

A line in R^3 can be described by a parametric equation of the form

[tex]\textbf{r}(t) = \textbf{r}_0+t\textbf{v},[/tex]

where [itex]\textbf{r}_0[/itex] is the position vector representing a point in [itex]\mathbb{R}^3[/itex], [itex]t[/itex] is a real number, and [itex]\textbf{v}[/itex] is a non-zero displacement vector indicating the direction of the line (and also its orientation: which way along the line is positive).

The condition for this line to be parallel to the xy-plane is that the z-component of [itex]\textbf{v}[/itex] is zero. That is, [itex]\textbf{v}[/itex] must be of the form (v₁,v₂,0), where v₁ and v₂ are fixed real numbers, not both 0. Suppose this is the case. If, and only if, v₁ is 0, the line will be parallel to the y-axis. If, and only if, v₂ is 0, the line will be parallel to the x-axis.

 

[kerrwilk]

Thanks! That was a great explanation.