Tangent line to a curve

The tangent line to a smooth curve $\mathbf{r}(t)$ at a point $\mathbf{r}(t_0)$ is the straight line through that point with direction ${\mathbf{r}}^{\prime }(t_0)$.

Parameterizing the tangent line by a new parameter $u$:

$\mathbf{L}(u)=\mathbf{r}(t_0)+u{\mathbf{r}}^{\prime }(t_0).$

At $u=0$ you’re at the contact point $\mathbf{r}(t_0)$. For each unit increase in $u$ you move by the tangent vector ${\mathbf{r}}^{\prime }(t_0)$.

Smoothness requirement

The construction requires ${\mathbf{r}}^{\prime }(t_0)\ne 0$. Otherwise no tangent direction exists — the curve has stalled or has a corner at $t_0$. This is part of why smooth curves are defined to have a nowhere-zero derivative.

Worked example

Tangent line to $\mathbf{r}(t)=⟨t{e}^{-t},t{e}^{-t},0⟩$ at $t_0=0$.

$\mathbf{r}(0)=⟨0,0,0⟩$. Differentiate: $(t{e}^{-t}{)}^{\prime }=(1-t)e^{-t}$, so ${\mathbf{r}}^{\prime }(t)=⟨(1-t)e^{-t},(1-t)e^{-t},0⟩$, and ${\mathbf{r}}^{\prime }(0)=⟨1,1,0⟩$.

Tangent line: $\mathbf{L}(u)=⟨u,u,0⟩$ — the line $y=x$, $z=0$.

Linear approximation

Near $t_0$, $\mathbf{r}(t)\approx \mathbf{r}(t_0)+(t-t_0){\mathbf{r}}^{\prime }(t_0)$. The tangent line is the first-order Taylor approximation to the curve.

This is how physicists treat smooth curves locally as straight lines, and how numerical algorithms (Euler’s method, linearization of ODEs around equilibria) work. See Locally linear system.

Tangent line vs. tangent vector

A subtle but useful distinction:

• Tangent vector ${\mathbf{r}}^{\prime }(t_0)$: an arrow, length-bearing, attached at $\mathbf{r}(t_0)$. Its magnitude is the speed.
• Tangent line $\mathbf{L}(u)$: a geometric line in space, infinite in extent, that contains the tangent vector.
• Unit tangent [[Unit tangent vector|$\mathbf{T}(t_0)$]]: the tangent vector normalized to length 1 — direction only, no speed information.

For line integrals the tangent vector is what you use; for visualizing the curve geometrically, the tangent line is what you draw.