Distance between 2 3D points in space. (proof)

[Curious_individual]

Hello everyone! I know how to calculate this, but I really struggle to get an intuition for why we do certain things. I understand the distance between 2D points proof. I just have trouble with distance between two 3D points. I provided an image which I was really confused by. We know the distance of P1 and B (x2 - x1)^2 + (y2 - y1)^2.
Let's assume for the rest of this that distance between P1 and B is 10. when we start calculating the distance of P1 and P2 we use pythagoras theorem because we have right angle. Why in the world don't we square the distance between P1 and B, but rather we do "Distance between P1 and P2 = squareroot(10+(z2-z1)^2)

Notes: I mean square with x^2

 

[blamocur]

...but rather we do "Distance between P1 and P2 = squareroot(10+(z2-z1)^2)

Where does this come from? I don't see this in your picture.

 

[Dr.Peterson]

We know the distance of P1 and B (x2 - x1)^2 + (y2 - y1)^2.

You mean, the square of the distance, right?

Why in the world don't we square the distance between P1 and B, but rather we do "Distance between P1 and P2 = squareroot(10+(z2-z1)^2)

If you are referring to this,



they are squaring that distance; what's in green is the square of P₁B. That would be 100 in your example, not 10.