Jacobian of a projection onto a plane Let $S$ be a surface orientable with smooth boundary in $\mathbb{R}^3$. Let $\Pi$ be a vector plane in $\mathbb{R}^3$ and define $f : S \to \Pi$ the orthogonal projection onto the plane. I computed the differential of this map
$$
(df)_p(w) = w - \langle w,v\rangle v, \quad \forall w \in T_p S
$$
where $v$ is the unit normal vector to $\Pi$. Let $n_S$ be the unit normal field to $S$. I want to prove that
$$
Jac(f)_p= |\langle v,n_s \rangle|, \quad \forall p \in S.
$$
I am using basis of $\Pi$ and $T_pS$ and I can't prove that... My formula was
$$
Jac(f)_p = \sqrt{\det\left(\delta_{ij} - \langle w_i,v\rangle \langle w_j,v\rangle \right)_{i,j=1}^2}
$$
where $\{w_1,w_2\}$ is a ortonormal basis of the $T_pS$. Anyone can help me?

HINT: Choose an orthonormal basis for the plane with one basis vector $w_1$ common to both planes. Think about vectors in the plane orthogonal to $w_1$.
