When tried to export any vector layer in ArcGIS [projected in UTM] to Google Earth, it is falling away from the actual location.
However, when I export vector layer in Geographical Coordinate System then it exactly falls in the location.
May I know the reason for this?
You are encountering the difference between a projected coordinate system and a geographic coordinate system.
Google Earth reads all coordinates as if they were in Geographic Coordinate System WGS 1984, a.k.a. WGS84. So latitude/longitude coordinates (45.215, -162.6548) will show up correctly (as long as you use the WGS84 datum).
Your UTM data, on the other hand, has a Projected Coordinate System, so its coordinates look like (5009510, 684145) which describes how many meters North and East the point is from the origin (0, 0) (basically the southwest corner) of whatever UTM Zone you are in. Exactly like a cartesian coordinate system in algebra.
This means that when Google Earth sees the coordinates (5009510, 684145) it will try to interpret them as degrees. You can expect funky results from trying to plot latitude = 5,009,510.00 degrees and longitude = 684,145.00 degrees.
To prepare data for Google Earth, use the Project tool to "project" (translate, recalculate, transpose, etc.) your UTM coordinates into GCS WGS84 coordinates.
Products like ArcMap and QGIS have the wherewithal to correctly display different coordinate systems on top of each other. This is generally called "projecting on-the-fly". Google Earth does not.
