Consider $f,g: S^{n}\rightarrow S^{n}$ to be continuous maps. Seeing $S^{n}\subset\mathbb{R}^{n+1}$, let's say that $f$ and $g$ are orthogonal at $x\in S^{n}$ whenever $\langle f(x),g(x)\rangle = 0$ for the usual inner product in $\mathbb{R}^{n+1}$.
How can I see that if $|deg(f)|\neq |deg(g)|$ then there is some point $x$ on which $f$ and $g$ are orthogonal ?
Here, $deg$ denotes the degree of a map.
Thank you
Suppose that there is no point where $f$ and $g$ are orthogonal. Then the inner product of $f$ and $g$ pointwise is always positive or always negative. In the first case, $g$ may be continuously deformed to $f$, just by moving $g(x)$ towards $f(x)$ in the arc joining them, which is the smaller of the two arcs in the great circle containing $f(x)$ and $g(x)$. In the second case, $g$ may be deformed to $-f$. But in any case, since the degree is preserved by continuous deformations, we would have $deg(g)=deg(\pm f)$, which is a contradiction.
