Tangent space of manifold has two unit vectors orthogonal to tangent space of its boundary

Question

I'm reading spivak calculus on manifolds and got stuck. Let M be a k-dimensional manifold with boundary in $\mathbb{R^{n}}$, and $M_{x}$ is the tangent space of M at x with dimension k, then $\partial M_{x}$ is the (k-1) dimensional subspace of $M_{x}$ Spivak says that this implies, there are only two unit vectors in $M_{x}$ that are orthogonal to $\partial M_{x}$. I am confused as to why only two unit vectors?

Answer 1

This is a linear algebra statement.

Given a finite dimensional inner product space $V$ and a codimension 1 subspace (that is, a hyperplane) $W \subset V$, there are precisely two unit vectors othogonal to $W$. Because $W$ is codimension 1, we automatically have that $\text{dim}(W^\perp) = 1$; and a 1-dimensional inner product space has precisely two unit vectors.