Distance between two parallel vector planes
How do I find the shortest distance between two planes? I've seen many formulas but I don't really understand how they work. Your vectors summary mentioned something about appending a plus sign between the two moduli if k1and k2 have different signs, and a minus sign if k1and k2 have the same signs. I'm not sure why this is so. o.O
Student X
When k1and k2 are of the same sign (either both positive or negative), then it means that both planes exist on the same side of the origin. As such, to compute the distance between them, it involves a subtraction of the separate distances from the two planes to the origin. If one is positive while the other is negative, then both planes happen on opposite sides of the origin; consequently the distance between them is achieved by adding them.
Perhaps this in depth explanation provided on my main site would clear things up:
http://www.whitegroupmaths.com/2010/01/understanding-matters-4.html
Hope this helps. Peace.
Best Regards,
Mr Koh
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Ah thanks! Is it possible to change the sign of k to make things simpler? Like, rewriting x -3y-2z = -4 as -x +3y+2z = 4. I think that'll give a different answer though.
Student X
It won't make any difference even if you change signs. An example would be able to sufficiently demonstrate this:
Eg. Find the distance between the planes x -3y -2z = -4 and -x +3y + 2z = 3
So x - 3y - 2z = -4 -----------(1)
and -x +3y + 2z = 3------------------(2)
Notice that the normals to both planes (while parallel) are running in opposite directions. So you must either modify (1) to give -x+3y+2z= 4 or modify (2) to give x-3y-2z= -3. This thus gives rise to the realization that both planes are actually on the same side of the origin (and not on the opposite side as misleadingly suggested by the values -4 and 3 on the RHS of (1) and (2) ), therefore the distance between them is (4-3)/sqrt(14) =1/sqrt(14) units.
Bear in mind when we find distances between planes, we are finding distances between parallel planes, because the distance between two non parallel planes is simply zero. ( ie they will definitely intersect)
Hope this helps. Peace.
Best Regards,
Mr Koh