I have this problem
For each $ y=(y_1, \dots, y_{n+1})\in S^n $ and each $ k=1,2,\dots, n+1$ we build $$ F_k (y)= e_k -y_k \cdot y \in \mathbb{R}^{n+1} $$ Use the functions $ F_k $ to build vector fields $ Y_1,\dots, Y_{n+1} $ over the sphere $ S^n $ ($ e_i $ are the canonical basis)
I understand that the vectors $ F_k(y)$ is ortogonal to $ y $, but I don't know how I can generate a vector field with that family of vectors.
The value of $F_k$ at $y$ is the orthogonal projection of the vector $e_k$ to $y^\perp=T_yS^n$. Hence the map $F_k:S^n\to\mathbb R^{n+1}$ has the property that $F_k(y)\in T_yS^n$, thus defining a vector field on $S^n$. (In addition, the values of these vector fields span the tangent space in each point.)