Im stuck in this exercise, where I have to determine two polynomials $p,q \neq 0$ that are orthogonal in regards to the inner product, where $i^2=-1$:
$$ \langle p,q \rangle = p(i) \cdot \overline{q(i)} + p(1) \cdot q(1) + \overline{p(i)} \cdot q(i) $$
$p,q \in P_3\mathbb{(R)}$ and have degree $\leq2$
$p(\alpha)=c_0+c_1\alpha+c_2\alpha^2$
$q(\beta)=d_0+d_1\beta+d_2\beta^2$
I know that the definition of orthogonality is
$\langle p,q \rangle=0$
I don't know how to solve this exercise. Do I have to solve the inner product equal to zero and isolate for $p$ and then $q$?
Or do I have to do something else?
Thanks in advance