How would you use Mathematica to verify the parallelogram law in real numbers^3 for any pair of vectors u and v.
In two dimensions:
v1 = {a, b}; v2 = {c, d} Expand[2 Tr[v1^2] + 2 Tr[v2^2] == Tr[(v1 + v2)^2] + Tr[(v1 - v2)^2]]
In three dimensions:
v1 = {a, b, c}; v2 = {d, e, f}; Expand[2 Tr[v1^2] + 2 Tr[v2^2] == Tr[(v1 + v2)^2] + Tr[(v1 - v2)^2]]
