Assuming that we have three-dimensional real vectors :
$Assumptions = (u | v | w) ∈ Vectors[3, Reals];
we can use e.g. various tensor functions (new in ver. 9)
e.g. TensorReduce to reduce (simplify) a tensor expression, e.g.
TensorReduce[v.v + w.w - (v + w).(v + w)]
TensorReduce[u\[Cross](v\[Cross]w)]
-2 v.w
-w u.v + v u.w
We can do more interesting reductions, let's show e.g. the Jacobi identity:
TensorReduce[ u\[Cross](v\[Cross]w) + v\[Cross](w\[Cross]u) + w\[Cross](u\[Cross]v)]
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