What's the formula for finding the shortest distance between two points in 3D?

[Patrick Kale]

Hi,

Given the question of finding the shortest distance between two points in 3D (xyz coordinates for each point), I was able to find the distance by basically imagining the given points as the vertices of a right triangle. Really easy problem to you guys but it was like a new thing to me and I used only my geometry knowledge instead of using a formula I did not know.

So basically I am just happy that I did this.

 

[symbolipoint]

Without trying to work through the description myself, the distance formula would at least give the actual size between the two points in three dimensions.

 

[Mentallic]

Hi,

Given the question of finding the shortest distance between two points in 3D (xyz coordinates for each point), I was able to find the distance by basically imagining the given points as the vertices of a right triangle. Really easy problem to you guys but it was like a new thing to me and I used only my geometry knowledge instead of using a formula I did not know.

So basically I am just happy that I did this.

I'd call it an application of Pythagoras' theorem.

 

[tahayassen]

Isn't this pretty much the only way of finding the distance between two points in 3D? Normally you would just use an extension of pythagorean's theorem.

 

[Patrick Kale]

Thanks for the replies guys. Application of Pythagorean's theorem is the consensus. I will do another attempt to find the distance formula for two xyz 3d points myself. I saw the formula after answering the first question then tried one time to make it myself but did not succeed.

 

[Best Pokemon]

It's called the Euclidean distance or Euclidean metric.

http://mathworld.wolfram.com/EuclideanMetric.html

http://en.wikipedia.org/wiki/Euclidean_distance

 

[Diffy]

Hi Patrick,

Good job on figuring this out for yourself. A lot of the mathematicians here will tell you that trying to figure things like this out on your own is a great way to learn.

I encourage you to keep at it.

 

[bp_psy]

Isn't this pretty much the only way of finding the distance between two points in 3D? Normally you would just use an extension of pythagorean's theorem.

You can also do it using a variational approach which I think is conceptually more clear since it specifically selects the path with the smallest length from all possible paths.

http://en.wikipedia.org/wiki/Distance#Variational_formulation_of_distance

 

[micromass]

Thanks for the replies guys. Application of Pythagorean's theorem is the consensus. I will do another attempt to find the distance formula for two xyz 3d points myself. I saw the formula after answering the first question then tried one time to make it myself but did not succeed.

Very well done man!

So, you have the formula for the distance of two points in 1D, in 2D and in 3D. Can you guess what the formula for the distance is in 4D, 5D?? Or in general: in n dimensions??

 

[Patrick Kale]

I produced the formula for the shortest distance between two points in the xy plane. It was a fun exercise. Here is a picture of the work I did:

http://postimage.org/image/b7tz5pvnp/full/

Users Diffy and micromass, thank you! User micromass, that seems like an interesting question. I will first do the attempt at the formula of distance between two points in the xyz plane and maybe I will be able to picture, or guess, an idea for the 4d and etc.