Does the unit tangent space of a point $p$ on the two-sphere, $S^2$ trace a unit circle?

Question

Let $p \in S^2 = \{(x,y,z)\in\mathbb{R}^3 : x^2 + y^2 + z^2 = 1 \}$. Let $v$ be a unit tangent vector at $p$. The unit tangent space, $UTS^2$ consists of all unit tangent vectors $v$ of $S^2$ at $p$.

Do the heads of each vector trace out a unit circle with center at $p$?