What are some features of the equations of a line that is parallel to the xz plane, but does not lie on the plane, and is not parallel to any of the axes?
So far all I got:
-dot product of plane's normal vector and the direction vector of the line must equal zero as the line is perpendicular to the normal vector
-line doesn't lie on plane, meaning one must choose a point that's not on that plane
I just don't know what features are needed to make sure the equation of a line isn't parallel with any of the axes. I feel like it's on the tip of my tongue but I can't quite grasp it.
Any ideas? Also, any other point you may come up with?
$\endgroup$ 51 Answer
$\begingroup$Every line parallel to the xz axis has the form $r(t) = (x_0,y_0,z_0) + (a,0,c)t$. Note that, $\vec{v}\cdot \vec{j} = (a,0,c)\cdot (0,1,0) = 0$. Ie, the vector $j$ is orthogonal director vector $r$. Therefore, $r$ is parallel to the $xz$ plane.
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