There are two formulas that are important to remember when considering vectors or positions in the 3D coordinate System. The midpoint formula and the distance formula in 3D. The midpoint and distance formula in 3D can be derived using a method of addition of the geometric representation of vectors. In order to understand the derivation of the distance formula in 3D we must understand 3D vector operations.
I want to derive the midpoint formula for 3 dimensions, the midpoint formula is going to help me find the midpoint between points a which is coordinates x1, y1 and z1 and b which has coordinates x2, y2 and z2. So I have segment ab drawn here and I've labeled my midpoint m and I'm hoping to find the formula for it's coordinates. I've also added a position vector oa for point a and a position vector om for point m. Now let's find the components for position vector oa, and let's recall that the components of a position vector are exactly the coordinates of the endpoint of that vector so there are going to be x1, y1, z1 and I'm also going to need vector ab in order to find m and what are the components of ab? Well since vector ab goes from point a to point b and the components are x2-x1, y2-y1 and z2-z1 okay how are we going to get position vector om from oa and ab? Well let's make the observation that the vector that starts at point a and ends at m is half of the vector that goes from a to be, so this is vector ab this vector starting here and ending here is a half of ab the scalar of multiple ohe half of ab and so I need to add that to oa to get om. So vector om=oa plus one half a b, so that's going to be in components x1, y1, z1 plus one half of this, one half of x2-x1, y2-y1 and z2-z1. So let's see if we can combine this in a single step for the first component I'm going to get x1 plus a half x2 minus a half x1 so a half x1 plus a half x2 now I'll get y1 plus a half y2 minus a half y1 that's one half y1 plus one half y2 and similarly I get one half z1 plus one half z2. Each of these is exactly the average of the x and y components of these 2 points, so I can write it as x1+x2 over 2 y1+y2 over 2 and z1+z2 over 2, these are the components of vector om which goes from the origin to point m and therefore the coordinates of point m are these. So the midpoint m of the segment joining x1, y1, z1 and x2, y2, z2 are x1+x2 over 2 y1+y2 over 2 and z1+z2 over 2 and that's the midpoint formula.