polarization identity
If you only know the squared lengths of vectors, can you recover the dot product that produced them? Surprisingly, yes — and the polarization identity is the formula that does it. It rebuilds a two-input symmetric pairing out of its single-input ‘squaring’ function, much as you can recover ab from (a+b)^2, a^2 and b^2 in ordinary algebra.
Given a quadratic form Q with polar form B(u, v) = Q(u + v) - Q(u) - Q(v), the symmetric bilinear form recovered from Q is B(u, v) = (1/2)[Q(u + v) - Q(u) - Q(v)], valid whenever 2 is invertible in the field. Equivalently B(u, v) = (1/4)[Q(u + v) - Q(u - v)]. Either way, the form is fully determined by its values Q(v) = B(v, v) on single vectors.
The hidden hypothesis is that 2 must be invertible. Over a field of characteristic 2 the factor 1/2 has no meaning and polarization fails: distinct quadratic forms can share the same polar bilinear form, which is exactly why quadratic forms are taken as primitive there. Polarization is therefore the bridge that makes ‘symmetric bilinear form’ and ‘quadratic form’ interchangeable away from characteristic 2.
For Q(x) = |x|^2 on R^n, polarization gives the dot product: (1/4)(|x+y|^2 - |x-y|^2) = x · y. Try x = (1,0), y = (0,1): (1/4)(2 - 2) = 0, correctly the orthogonality of e_1 and e_2.
Recovering the dot product from the squared norm.
The same trick appears in inner product spaces, where it recovers the inner product from the norm, and in higher degree as the formula expressing a symmetric multilinear map from its diagonal restriction.