tangential and normal projection of a vector in the ambient vector field of a sphere

I'm having unexpected trouble to perform this computation:

Let $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2=3\}$ and $v_p = (1,0,0)_{(1,1,1)}$ be a vector from the ambient vector field on $M$. How do I now compute the projections $v_p^T$ of $v_p$ to the tangent space of $M$ at $p$ and $v_p^N$ to the normal space of $M$ at $p$. I am feeling bad about my disability to solve this task, because it seems to be basic and easy. Explanation of steps welcome!

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At $(1,1,1)$ the unit normal vector is $n = \frac{1}{\sqrt 3}(1,1,1)$. The projection of $v$ onto $n$ is $$v^N = (v \cdot n) n = \frac{1}{3}(1,1,1).$$ Now since $v^T + v^N = v$ we can solve $$v^T = v - v^N = (2/3, -1/3,-1/3).$$