I have a couple of questions to answer, and I am unsure if i argue correctly:
Given two vectors $a$, $b$ with only strictly positive coordinates, can those two vectors be orthogonal? My answer would be no. As $\langle a,b\rangle=0$ this can only be the case if all $ab$-coordinates are $0$, which is not the case because the coordinates have to be strictly positive, so in ordered to get to $0$ some ab products have to be negative.
Can $ca$ and $db$ be orthogonal, for $c$,$d$ elements of $\mathbb R$?
No if $a$ and $b$ are not orthogonal? I'm not sure if this question refers to the question above....my second question would be if $c$ and $d$ are $0$, $\langle ca,db\rangle$ would also be $0$ would this be than count as orthogonal??
How many vectors build a orthonormal basis in $\mathbb R^n$ - n
How many of these vectors (of the basis above) can have strictly positive Coordinates, how many strictly negative?
I would guess only $1$, because if all vectors are orthogonal than there can be only one with strictly positive coordinates..
Many thanks for your help!!