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Posted by White Group Mathematics on March 23, 2012 at 12:05 PM |

How do I find the shortest distance between two planes? I've seen many formulas but I don't really understand how they work. Whitecorp, your vector's summary said something about putting a minus sign between the two modulus' if k1 and k2 have different signs. And about putting a plus sign if k1 and k2 have the same signs. I'm not sure how that works. o.O

Student X

When k1 and k2 are of the same sign ( either both positive or negative), then it means that both planes are on the same side of the origin.Hence, to find the distance between the two planes, it is a subtraction of the separate distances of the two planes to the origin. If one is positive while the other is negative, then the two planes are each on the opposite side of the origin; hence the distance between them is achieved by adding them.

Perhaps this in depth explanation on my main site would help:

**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, x - 3y - 2z = -4 change to -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 make things clearer:

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. Soyou either modify (1) to give -x+3y+2z= 4 or modify (2) to give x-3y-2z= -3. This creates the realization that both planes are actually **on the same side of the origin** (and not on the opposite side as misleadingly given by the values -4 and 3 on the RHS of (1) and (2) ), and 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

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