Unit tangent vector

The unit tangent vector to a smooth curve $\mathbf{r}(t)$ is

$\mathbf{T}(t)=\frac{{\mathbf{r}}^{\prime }(t)}{|{\mathbf{r}}^{\prime }(t)|}.$

It points in the direction of motion along the curve and has magnitude $1$. The separation of “direction” ($\mathbf{T}$) from “speed” ($|{\mathbf{r}}^{\prime }|$) is what makes the unit tangent useful.

Why a unit vector

[[Derivative of vector-valued function|${\mathbf{r}}^{\prime }(t)$]] gives a tangent direction but its magnitude depends on the parameterization. Two parameterizations of the same curve (one slow, one fast) have very different ${\mathbf{r}}^{\prime }$ magnitudes at corresponding points, even though geometrically the curve has the same direction. Dividing by $|{\mathbf{r}}^{\prime }|$ kills the parameterization dependence and isolates pure direction.

In arc-length parameterization (where $|{\mathbf{r}}^{\prime }(s)|=1$ for all $s$), $\mathbf{T}={\mathbf{r}}^{\prime }$ — the derivative is already a unit vector.

In line integrals

The scalar line integral of a vector field along a curve can be written

${\int }_C\mathbf{F}\cdot d\mathbf{r}={\int }_C\mathbf{F}\cdot \mathbf{T}ds,$

where $ds=|{\mathbf{r}}^{\prime }(t)|dt$ is the arc-length element. The integrand $\mathbf{F}\cdot \mathbf{T}$ extracts the component of the field along the direction of motion — only the parallel component contributes; perpendicular components do no work. This is the “work” interpretation of $\int \mathbf{F}\cdot d\mathbf{r}$.

Frenet frame (preview)

Beyond $\mathbf{T}$, two more unit vectors complete the Frenet frame: the principal normal $\mathbf{N}={\mathbf{T}}^{\prime }/|{\mathbf{T}}^{\prime }|$ (direction the curve is bending) and the binormal $\mathbf{B}=\mathbf{T}\times \mathbf{N}$ (perpendicular to the osculating plane). These define curvature and torsion of space curves. Vector Calculus and Complex Analysis doesn’t develop this; the unit tangent alone suffices for line integrals.

Examples

Straight line $\mathbf{r}(t)={\mathbf{P}}_0+t\mathbf{v}$: ${\mathbf{r}}^{\prime }(t)=\mathbf{v}$, $\mathbf{T}=\mathbf{v}/|\mathbf{v}|$, constant.

Helix $\mathbf{r}(t)=⟨\cos t,\sin t,t⟩$: ${\mathbf{r}}^{\prime }=⟨-\sin t,\cos t,1⟩$, $|{\mathbf{r}}^{\prime }|=\sqrt{2}$, $\mathbf{T}=⟨-\sin t,\cos t,1⟩/\sqrt{2}$. Constant magnitude but rotating direction.