The mysterious 4th dimension

Imagine we have a 2D plane-what would be the axis of the next,third dimension?Let's draw a line perpendicular to that plane.It's obvious that this line would be also perpendicular to every possible line in that plane,and also It's parallel to the third axis.
Now-wouldn't the axis of the 4th dimension have to be perpendicular to every possible line in 3D space?Is my reasoning correct,/sci/?

If time is the 4th dimension,then how fast am I moving sitting at my computer?1 second/second=1?
Oh,and according to Lorentz contraction (or something),If I move faster in the 3 spatial dimensions,i move slower in the 4th-how the hell does that work?if I increase velocity in the z axis,the x and y velocities won't decrease,but w decreases?How the hell would you work with vectors then in this framework?