The code you have requires loading the old package. The built-in vector analysis functions have different names or different ways to obtain the same output. See http://reference.wolfram.com/language/Compatibility/tutorial/VectorAnalysis.html or its built-in equivalent, Compatibility/tutorial/VectorAnalysis under Help - Wolfram Documentation.
The above tutorial suggests replacing DotProduct from the old package with a user function that converts the vectors to Cartesian, then takes the ordinary Dot product.
In[1]:= dotProduct[a_, b_, chart_] :=
CoordinateTransform[chart -> "Cartesian", a].CoordinateTransform[ chart -> "Cartesian", b]
In[2]:= dotProduct[{3, 2, 4}, {3, 2, 4}, "Cylindrical"] // N
Out[2]= 25.
Using the package works for me with 10.0.0 and 10.0.1.
In[1]:= Needs["VectorAnalysis`"]
General::obspkg: VectorAnalysis` is now obsolete. The legacy version being loaded may conflict with
current functionality. See the Compatibility Guide for updating information.
In[2]:= DotProduct[{3, 2, 4}, {3, 2, 4}, Cylindrical]
2 2
Out[2]= 16 + 9 Cos[2] + 9 Sin[2]
In[3]:= Simplify[%]
Out[3]= 25
In[4]:= $Version
Out[4]= 10.0 for Microsoft Windows (64-bit) (June 29, 2014)
The attached notebook has three examples - your code without loading the package, your code after loading the package, and the new method.
Attachments: