Orthogonal vectors in the positive cone of $\Bbb R^m$

Question

What is the meaning of the statement " $v_1, v_2, \dots, v_n$ are orthogonal vectors in the positive cone of $\Bbb R^m$". I have encountered that in a book.

Answer 1

Most likely we're looking at the set $$C = \{ {\bf x} \in \Bbb R^m \mid x_1^2+\cdots+ x_{m-1}^2 = x^2_m \ \mbox{ and } \ x_m >0 \},$$and we're taking ${\bf v}_1,\cdots,{\bf v}_n \in C$ (we can pick these vectors being orthogonal to each other if $n \leq m$, just intersect $C$ with convenient hyperplanes).

Think about it in $\Bbb R^3$ first to gain intuition.