I'm not sure how to draw a line that intersects all three coordinate planes. I'm suspicious that such a line doesn't exist. How do we prove its existence? I guess we should try draw a picture. I beg for a picture.
2026-03-28 16:45:50.1774716350
Visualization/Drawing Request of a line L intersecting all three coordinate planes Oxy, Oyz, and Ozx
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All the lines that are not parallel to a coordinate plane intersect all the three coordinate planes.
In the figure you can see the line that passes thorough the points $A=(2,1,0)$ and $ B=(0,2,1)$, that has equation $(x,y,z)=t(2,-1,-1)+(0,2,1)$ and intesects the plane $y=0$ at $C=(4,0,-1)$.