= Unit tangent vector
{title2=$T$}
For a <regular curve> $\mathbf r(t)$ with nonzero <velocity vector>, its unit tangent vector is
$$
T(t)=\frac{\mathbf r'(t)}{\|\mathbf r'(t)\|}.
$$
The <Euclidean norm> in the denominator removes the traversal <speed>, leaving a <unit vector> that records direction. A positive change of traversal <speed> leaves this direction unchanged; reversing the direction of travel reverses $T$. In <arc length> coordinates, $T=d\mathbf r/ds$.
For the <parabola> $\mathbf r(t)=(t,t^2)$, the <velocity vector> is $(1,2t)$ and the <speed> is $\sqrt{1+4t^2}$. At $t=1$, $T=(1,2)^T/\sqrt5$. The <tangent vector> $(1,2)^T$ and the <unit tangent vector> point in the same direction, but only the latter has <Euclidean norm> one. The <principal normal vector> is a further choice based on how this direction changes, where the <curvature> is nonzero. https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature[OpenStax Calculus Volume 3, §3.3: unit tangent and principal normal]{ref}
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