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.
If a line in ๐ ยณ (three-dimensional Cartesian coordinate system) is parallel to the xy-plane but not to any of the axes, it means that the line lies entirely within the xy-plane and doesn't intersect any of the coordinate axes (x-axis, y-axis, or z-axis). In other words, it has no component in the z-direction.
To determine the equations of such a line, you can use a parametric equation or a symmetric equation.
To find the parametric equations, we'll start by considering a point on the line, let's call it ๐(๐ฅโ, ๐ฆโ, ๐งโ). Since the line is parallel to the xy-plane and has no component in the z-direction, we know that ๐ง = ๐งโ for all points on the line.
Now, let's denote ๐ก as a parameter that varies over the real numbers. From the point ๐(๐ฅโ, ๐ฆโ, ๐งโ), we can determine any other point on the line by using vector notation:
๐ = ๐ + ๐ก๐ฃ,
where ๐ is the position vector of any point on the line, ๐ = (๐ฅโ, ๐ฆโ, ๐งโ) is the position vector of the initial point, ๐ก is the parameter, and ๐ฃ is the direction vector of the line.
Since the line is parallel to the xy-plane, we can choose the direction vector ๐ฃ to have components (a, b, 0), where a and b are not both zero. This ensures that the line doesn't lie on any of the coordinate axes.
So, the parametric equations of the line parallel to the xy-plane but not to any of the axes are:
๐ฅ = ๐ฅโ + ๐ก๐,
๐ฆ = ๐ฆโ + ๐ก๐,
๐ง = ๐งโ,
where ๐ and ๐ can be any real numbers, not both zero.
This set of equations represents all the points on the line as ๐ก varies over the real numbers.
I hope this explanation helps you understand how to derive the parametric equations for a line parallel to the xy-plane but not parallel to any of the axes. Let me know if you have any further questions!