eStoY confUSo
Why didn't he use the dot product?
I'm hoping that $\mathbb P_2$ is a polynomial space (I don't even know what that means), probably representable as a vector in $\mathbb R^3$.
So now that I'm already so confused I want to make a LITTLE bit of sense here. Two vectors are orthogonal in this vector space if their inner product is zero? Because what he did accomplishes that.
Best Answer
You're right, the inner product of two polynomials is defined as $$\langle\, p,q\, \rangle=\int_0^1p(t)q(t)\,\mathrm dt, $$ and two polynomials are orthogonal if and only if $\;\langle\, p,q\,\rangle=0$.
These definitions, b.t.w. are valid in the (infinite-dimensional) vector space $\mathbf R[t]$ of all polynomials, and not only in the vector space of degree at most $2$.
He didn't use the dot product of the polynomial coefficients (if I understand well what you mean) simply because the ordinary dot product isn't interesting here, and he chooses another, more useful, inner product.