Question about the definition of a vector field on an $n$-sphere

Question

From Rotman's algebraic topology:

A vector field on $S^m$ is a continuous map $f : S^m \rightarrow \Bbb R^{m+1}$ with $f(x)$ tangent to $S^m$ at $x$ for every $x \in S^m$.

I don't understand what "with $f(x)$ tangent to $S^m$ at $x$ for every $x \in S^m$" means.

If you take $m=1$, then it's a continuous map from $S^1 \rightarrow \Bbb R^2$, but what does that phrase mean with respect to this or any higher $m$? Does it mean the line through $x \in S^m$ and $f(x) \in \Bbb R^{m+1}$ is tangent to $S^m$?

Answer 1

It means the line through $x$ in the vector direction $f(x)$ (that is, the line through $x$ and $x+f(x)$) is tangent to $S^m$. More precisely, it means that there exists a smooth path $\gamma:\mathbb{R}\to \mathbb{R}^{m+1}$ whose image is contained in $S^m$ such that $\gamma(0)=x$ and $\gamma'(0)=f(x)$. Concretely, this is equivalent to saying the vectors $x$ and $f(x)$ are orthogonal.

Answer 2

It means that $f(x)$ is orthogonal to $x$ with respect to the canonical inner product.