Orthogonality Condition

Question

Consider the set $$ Z_n = \lbrace (x,y) \in \mathbb{R}^n \times \mathbb{R}^n \; | \; x^\top y = 0 \rbrace. $$ It is easy to see that $$ Z_1 = \lbrace (x,0) \in \mathbb{R}^2 \rbrace \cup \lbrace (0,y) \in \mathbb{R}^2 \rbrace $$ is contained in a finite union of hyperplanes. Does the same hold true for $n \geq 2$?

Answer 1

$$f_x(y):y\in \mathbb{R}^n\to x^Ty$$ so $Z_n=\{(x,f_x^{-1}(0)), x\in \mathbb{R}^n\}=\{(0,y), y\in \mathbb{R}^n\} \cup \{(x,f_x^{-1}(0)), x\neq 0,x\in \mathbb{R}^n \}$ since for any $x$ one can construct non zero $y$ with $x^Ty=0$ and vise versa, this is the form of $Z_n$.