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

$$\textbf{r}(t) = \textbf{r}_0+t\textbf{v},$$

where $\textbf{r}_0$ is the position vector representing a point in $\mathbb{R}^3$, $t$ is a real number, and $\textbf{v}$ 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 $\textbf{v}$ is zero. That is, $\textbf{v}$ 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.