Why is $\langle p,q \rangle = 0$?

Question

First of all, sorry for opening a new question about it, but I'm curious to understand:

John Hughes claims that $\langle p,q \rangle =0$ (in the end of his answer)

Why is it true?

Prove that there is a unique inner product on V

Answer 1

If $p \in W_1, q \in W_2$, then write $p = p_1 + 0, q = 0 + q_2$ so $\langle p, q \rangle = \langle p_1,0\rangle_1 + \langle 0,q_2\rangle_2 = 0$.