On the page on equations of planes from Paul's Online Math notes, there's this statement;
Also notice that we put the normal vector on the plane, but there is actually no reason to expect this to be the case. We put it here to illustrate the point. It is completely possible that the normal vector does not touch the plane in any way.
How exactly does this work out? The only way I can interpret what he said is 'since you can move vectors around without changing their direction, you can translate the vector $\hat{n}$ so that it moves out of the plane', but that seems trivial; there isn't any reason for him saying that there.
Which cases are being referred to here, where a vector normal to a plane doesn't come in contact with it?
Best Answer
Since the excerpt is from an introductory section about equations of lines and planes, the author is indeed just emphasising/reminding that unlike points, lines and planes—which are fixed in space— $\vec n,$ being a vector, is “portable” by translation in space.