I think the reason is that because the basic structure of the computer languages is in binaries (numbers), the data capturing medium (remote sensing technology) also digital (0 and 1 - i.e numbers ); at the same time, the statistical/mathematical methods built into ArcGIS for spatial analysis are also based on numbers. Hence such ArcGIS tools that are specifically for raster data analysis may not work on a vector data except it is brought to a raster format.

Hayford O. Ahiadu

The vectors are geometric features such as points, lines and polygons. The way which works spatial analysis is a mathematical elaboration of numerical data and only with raster it's possible represent the results, then you can transform the math spatial data of your interest in vectors such as topographic isocontours etc.

It is not necessary to conduct spatial analysis only with raster representations. Spatial analysis can be conducted on point, vector, polygon as well as raster representations; spatial analysis also can be conducted using different representations (e.g. by analyzing spatial and attribute relationships among point, polygon and raster representations simultaneously). The SpaceStat software does this, see www.biomedware.com

The question is about ArcGis and the tool named Spatial analyst, as Barath says, perfoms analysis for example on attributes of the related vector file, the results then aren't a vector but a raster file. Obviousely it depends on the data and the analysis you need. The only way to obtained a vctor file is on utilized the "Srhortest path" if I remember. So then with the surface analyst you can obtained other vector. In many software the options such merge, dissolve, clip etc. are defined as Geoprocessing tools and you can work directly with vector files but the resulta are ever raster. This derives by my little work experience.

The Spatial Analyst extension is slightly inaccuretly named (probably due to the history of Arc development), Arc itself offers other options to do spatial analyses based on vector data. If we take the example of a digital terrain model, you can either use rasters or TINs. Really this extension should be called Raster Analysis (or Grid Analysis).