Inner product space and orthogonal complement

Question

This is the problem in my current homework:

"Let V be a $\mathbb{K}$-inner product space and let $U ⊆ V$ be a subset. We define the orthogonal complement $U^⊥$ of $U$ as $ U^⊥ := \{v ∈ V : ∀u ∈ U : \left\langle u, v \right\rangle = 0\}. $

For V = $\mathbb{C^3}$ equipped with the standard dot product calculate $\{z ∈ \mathbb{C^3}: z_1 + iz_2 + 2z_3 = 0\}^⊥$ "

I genuinely have no idea what I'm supposed to do here. Like... at all.

Answer 1

Use the fact that $z_1 + iz_2 + 2z_3=z_1\cdot 1 + z_2\cdot (-\bar{i}) + z_3\cdot 2=\left\langle (z_1,z_2,z_3),(1,-i,2) \right\rangle$ and you're done.

Answer 2

Hint: $ z_1+i z_2+2z_3 $ is a dot product...