How do we know the distance and the direction between two gps postions?

Question

please consider the problem of having different heights (hills mountains,,etc) between two postions (degrees and minutes)

Answer 1

We don't!  The satellites do!

Answer 2

A typical GPS receiver calculates its position using the signals from four or more GPS satellites. Four satellites are needed since the process needs a very accurate local time, more accurate than any normal clock can provide, so the receiver internally solves for time as well as position. In other words, the receiver uses four measurements to solve for four variables: x, y, z, and t. These values are then turned into more user-friendly forms, such as latitude/longitude or location on a map, then displayed to the user.

Each GPS satellite has an atomic clock, and continually transmits messages containing the current time at the start of the message, parameters to calculate the location of the satellite (the ephemeris), and the general system health (the almanac). The signals travel at the speed of light through outer space, and slightly slower through the atmosphere. The receiver uses the arrival time to compute the distance to each satellite, from which it determines the position of the receiver using geometry and trigonometry.

Although four satellites are required for normal operation, fewer may be needed in some special cases. If one variable is already known (for example, a sea-going ship knows its altitude is 0), a receiver can determine its position using only three satellites. Also, in practice, receivers use additional clues (doppler shift of satellite signals, last known position, dead reckoning, inertial navigation, and so on) to give degraded answers when fewer than four satellites are visible.