I have a set of raster datasets and have created a raster catalog out of them. These datasets should cover specific vector data. I have just realised that the raster as tif data have a coordinate system based on the efw file but it is not clear for me what is the coordinate system. I have to cover a vector data which has the coordinate of 4647 (EPSG) . I have tried to change it arc catalog based on the guidance from ESRI ( http://desktop.arcgis.com/en/arcmap/10.3/manage-data/raster-and-images/defining-or-modifying-a-raster-coordinate-system.htm) but it still place in a same position and practically there is no change. How it is possible to do it ?
You will have to figure out what coordinate system the images are in to start with. Once you know that, the first step is to use the process in the help page you cited to tell ArcGIS what it is. (In the ArcGIS help, where they talk about defining a coordinate system, you are not changing the image data, you are just informing ArcGIS what coordinate system it is in.) THEN you can use the projection tools to create a new version of the data in whatever coordinate system you want. (Or you might not have to, if the on-the-fly conversion in ArcMap suits you.) But ArcGIS can't help you until it knows what you're starting with.