MATH 261 — Multivariable Calculus

Lecture App: Vector-Valued Functions and Parametric Curves

Date: February 7, 2026

Instructor: Dr. V. S. Oussa

Mode: Board / Hybrid

Prereqs: Vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$, limits & derivatives, basic trig

Reading: Paul’s Online Notes (Parametric Equations & Vector Functions) or your class notes

0% complete · Stored locally in your browser

Motivation (particle motion). Think of a particle moving in space. At time $t$, its location can be encoded by a position vector $\mathbf{r}(t)$ whose tail is at the origin and whose head is at the particle.

Core definition. A vector-valued function is a function whose input is a real parameter and whose output is a vector (typically in $\mathbb{R}^2$ or $\mathbb{R}^3$).

Learning outcomes. By the end of this lecture, you should be able to:

Write vector-valued functions in component form and interpret them as parametric curves.
Compute the domain of $\mathbf{r}(t)$ as an intersection of component domains.
Recognize when $\mathbf{r}(t)=\mathbf{p}+t\mathbf{v}$ represents a line and identify $\mathbf{p}$ and $\mathbf{v}$.
Analyze and sketch the helix $\mathbf{r}(t)=\langle \cos t,\sin t,t\rangle$ and relate it to the cylinder $x^2+y^2=1$.
Differentiate and integrate vector-valued functions componentwise; compute tangent vectors and tangent lines.

How to use. Select a tile to switch videos. If a video is playing, switching tiles will stop the previous one automatically.

Vector-Valued Functions (Part 1)

Vector Calculus (Derivatives & Tangents)

Component form. A vector-valued function in $\mathbb{R}^n$ is written as

$$\mathbf{r}(t)=\langle f_1(t),f_2(t),\dots,f_n(t)\rangle.$$

The real-valued functions $f_1,\dots,f_n$ are called the component functions.

Parametric equations. In $\mathbb{R}^3$, writing $\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle$ induces the parametric system

$$x=x(t),\qquad y=y(t),\qquad z=z(t).$$

The curve traced by the terminal point of $\mathbf{r}(t)$ is a parametric curve.

Example. $\mathbf{r}(t)=\langle t^2,t^3\rangle$ defines a planar parametric curve. As $t$ changes, the point $(t^2,t^3)$ moves, tracing the curve.

Rule. The domain of $\mathbf{r}(t)=\langle f(t),g(t),h(t)\rangle$ is the intersection

$$\mathrm{Dom}(\mathbf{r})=\mathrm{Dom}(f)\cap \mathrm{Dom}(g)\cap \mathrm{Dom}(h).$$

Worked example. Let $$\mathbf{r}(t)=\langle \cos t,\ \ln(4-t),\ \sqrt{t+1}\rangle.$$

$\cos t$ is defined for all $t\in\mathbb{R}$.
$\ln(4-t)$ requires $4-t>0\Rightarrow t<4$.
$\sqrt{t+1}$ requires $t+1\ge 0\Rightarrow t\ge -1$.

Therefore $$\mathrm{Dom}(\mathbf{r})=\mathbb{R}\cap(-\infty,4)\cap[-1,\infty)=\boxed{[-1,4)}.$$

Use Tool A in the Interactive Lab to generate and solve many domain intersections quickly (with clear steps).

Lines. If $$\mathbf{r}(t)=\mathbf{p}+t\mathbf{v},$$ then the curve is a line through the point $\mathbf{p}$ in the direction $\mathbf{v}$.

Example. $$\mathbf{r}(t)=\langle 1+t,\ 2+5t,\ -1+6t\rangle =\langle 1,2,-1\rangle + t\langle 1,5,6\rangle.$$

The helix. Consider $$\mathbf{r}(t)=\langle \cos t,\ \sin t,\ t\rangle.$$

Because $\cos^2 t+\sin^2 t=1$, the curve lies on the cylinder $x^2+y^2=1$ and spirals upward as $z=t$ increases. This space curve is called a helix.

Derivative (definition). $$\mathbf{r}'(t)=\lim_{h\to 0}\frac{\mathbf{r}(t+h)-\mathbf{r}(t)}{h}.$$

Component rule. If $\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle$, then $$\mathbf{r}'(t)=\langle x'(t),y'(t),z'(t)\rangle.$$

Tangent line. The tangent line to $\mathbf{r}(t)$ at $t=t_0$ is

$$\mathbf{L}(s)=\mathbf{r}(t_0)+s\,\mathbf{r}'(t_0),\qquad s\in\mathbb{R}.$$

Helix at $t_0=\pi/2$. $\mathbf{r}(\pi/2)=\langle 0,1,\pi/2\rangle$ and $\mathbf{r}'(t)=\langle -\sin t,\cos t,1\rangle$, hence $\mathbf{r}'(\pi/2)=\langle -1,0,1\rangle$.

So one tangent line parametrization is $$x=-s,\quad y=1,\quad z=\pi/2+s.$$

Constant magnitude implies perpendicularity. If $\|\mathbf{r}(t)\|$ is constant, then $$\mathbf{r}(t)\cdot \mathbf{r}'(t)=0,$$ so $\mathbf{r}(t)\perp \mathbf{r}'(t)$. (Differentiate $\mathbf{r}\cdot\mathbf{r}$.)

Integral (definition). For $\mathbf{r}(t)=\langle f(t),g(t),h(t)\rangle$,

$$\int_a^b \mathbf{r}(t)\,dt=\left\langle \int_a^b f(t)\,dt,\ \int_a^b g(t)\,dt,\ \int_a^b h(t)\,dt\right\rangle.$$

Example. $$\int_0^{\pi/3}\langle 2\cos t,\ \sin t,\ 2t\rangle\,dt=\boxed{\langle \sqrt3,\ 1/2,\ \pi^2/9\rangle}.$$

Check 1. Explain (in one sentence) why the domain of a vector-valued function is an intersection of component domains.

Check 2. If $\mathbf{r}(t)=\langle \cos t,\sin t\rangle$, why is $\mathbf{r}(t)\perp \mathbf{r}'(t)$?

Check 3. For $\mathbf{r}(t)=\mathbf{p}+t\mathbf{v}$, what is $\mathbf{r}'(t)$? What does this mean geometrically?

Use the Practice tab for auto-checked versions of these checks.

Suggested practice set.

HW1 (Domain): Find the domain of $\mathbf{r}(t)=\langle \sin t,\ \ln(2+t),\ \sqrt{5-t}\rangle$.
HW2 (Line form): Rewrite $\mathbf{r}(t)=\langle 3-2t,\ 1+4t,\ 7+t\rangle$ as $\mathbf{p}+t\mathbf{v}$ and identify $\mathbf{p},\mathbf{v}$.
HW3 (Helix): Show that $\mathbf{r}(t)=\langle \cos t,\sin t,t\rangle$ lies on $x^2+y^2=1$. Compute $\mathbf{r}'(t)$.
HW4 (Tangent line): Find the tangent line to the helix at $t=\pi$.
HW5 (Integral): Compute $\int_0^1 \langle t,\ t^2,\ e^t\rangle\,dt$ componentwise.

Resources

Homework	MyLab
Paul’s Online Notes	Vector Functions Calculus with Vector Functions Parametric Equations
Mathematica reference	ParametricPlot ParametricPlot3D Integrate

Tip: In the Interactive Lab, the Helix Explorer works as a visual companion to 3D parametric plotting.

Paul Dawkins, Paul’s Online Notes, Calculus III: Vector Functions and Calculus with Vector Functions.
Any standard Calculus III text (e.g., Stewart): sections on parametric curves and vector-valued functions.

Interactive Lab. These tools make the computations concrete and the geometry visible. Everything runs locally in your browser.

Tool A — Domain Explorer (Intersection of Component Domains)

Mark complete

Include $\ln(a-t)$ (requires $t<a$)

Use log

Include $\sqrt{t+b}$ (requires $t\ge -b$)

Use sqrt

Optional: include $\dfrac{1}{\sqrt{c-t}}$ (requires $t<c$)

Use denom sqrt

Domain: —

Interpretation: overlap of all required conditions (intersection on the number line).

Show the reasoning step-by-step

Tool B — From $\mathbf{r}(t)=\langle a_1+b_1t,\ a_2+b_2t,\ a_3+b_3t\rangle$ to $\mathbf{p}+t\mathbf{v}$

Mark complete

Enter coefficients (constants $a_i$ and $t$-coefficients $b_i$)

Constants $(a_1,a_2,a_3)$

$t$-coefficients $(b_1,b_2,b_3)$

Point $\mathbf{p}$: —

Direction $\mathbf{v}$: —

Vector equation: —

Geometric meaning: the curve is a line through $\mathbf{p}$ in direction $\mathbf{v}$.

Show the decomposition

Conceptual check. For a line, $\mathbf{r}'(t)=\mathbf{v}$ is constant. This means the velocity is constant and points in the direction of motion.

Tool C — Helix Explorer: $\mathbf{r}(t)=\langle \cos t,\sin t,t\rangle$

Mark complete

Choose a parameter value $t$ (radians)

Current: —

$\mathbf{r}(t)$: —

$\mathbf{r}'(t)$: —

Cylinder check: $x^2+y^2=$ — (should be 1)

Show tangent line at this $t$

3D viewer not available. Enable show3d="1" on the shortcode, and verify that your browser permits WebGL.

3D visualization (helix + moving point + tangent direction)

Drag to rotate • Scroll to zoom • The tangent arrow shows the direction of $\mathbf{r}'(t)$.

Tool D — Curve Tracer (2D): $\mathbf{r}(t)=\langle t^2,\ t^3\rangle$

Mark complete

Choose $t_0$

Current $t_0$: —

Point $\mathbf{r}(t_0)$: —

Tangent vector $\mathbf{r}'(t_0)$: —

Tangent line (parametric): —

Show the derivatives

The curve is drawn in the $(x,y)$-plane; the highlighted point corresponds to $t_0$, and the tangent line is drawn at that point.

Tool E — Vector Integral (componentwise)

Mark complete

This tool matches the worked example: $\displaystyle \int_a^b \langle 2\cos t,\ \sin t,\ 2t\rangle\,dt$.

Lower bound $a$

Upper bound $b$

Value: —

Show componentwise steps

Practice. These checks are designed to build fluency: domains, tangent vectors/lines, and componentwise integration.

Formative Check 1 — Domain of a Vector-Valued Function

Mark complete

For $\mathbf{r}(t)=\langle \cos t,\ \ln(4-t),\ \sqrt{t+1}\rangle$, enter the domain as an interval.

Left bracket

Left endpoint

Right endpoint

Right bracket

Formative Check 2 — Tangent Vector on the Helix

Mark complete

Let $\mathbf{r}(t)=\langle \cos t,\ \sin t,\ t\rangle$. Compute the tangent vector $\mathbf{r}'(\pi/2)$.

x-component

y-component

z-component

Formative Check 3 — A Vector Integral

Mark complete

Compute $\displaystyle \int_0^{\pi/3}\langle 2\cos t,\ \sin t,\ 2t\rangle\,dt$ (enter numeric values; rounding is fine).

First component

Second component

Third component

Practice Generator

Mark complete

Topic

Show solution

Vector-valued function

Lecture App: Vector-Valued Functions and Parametric Curves

Purpose, Context, and Learning Outcomes

Lecture Videos

Definitions and Notation

Domains: the Intersection Principle

Geometry: Lines and the Helix

Calculus on Vector-Valued Functions

Formative Checks (low-stakes)

Homework, Reading, and Resources

Resources

References and Attribution

Tool A — Domain Explorer (Intersection of Component Domains)

Tool B — From $\mathbf{r}(t)=\langle a_1+b_1t,\ a_2+b_2t,\ a_3+b_3t\rangle$ to $\mathbf{p}+t\mathbf{v}$

Tool C — Helix Explorer: $\mathbf{r}(t)=\langle \cos t,\sin t,t\rangle$

Tool D — Curve Tracer (2D): $\mathbf{r}(t)=\langle t^2,\ t^3\rangle$

Tool E — Vector Integral (componentwise)

Formative Check 1 — Domain of a Vector-Valued Function

Formative Check 2 — Tangent Vector on the Helix

Formative Check 3 — A Vector Integral

Practice Generator

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Vector-valued function

Lecture App: Vector-Valued Functions and Parametric Curves

Resources

Tool A — Domain Explorer (Intersection of Component Domains)

Tool B — From $\mathbf{r}(t)=\langle a_1+b_1t,\ a_2+b_2t,\ a_3+b_3t\rangle$ to $\mathbf{p}+t\mathbf{v}$

Tool C — Helix Explorer: $\mathbf{r}(t)=\langle \cos t,\sin t,t\rangle$

Tool D — Curve Tracer (2D): $\mathbf{r}(t)=\langle t^2,\ t^3\rangle$

Tool E — Vector Integral (componentwise)

Formative Check 1 — Domain of a Vector-Valued Function

Formative Check 2 — Tangent Vector on the Helix

Formative Check 3 — A Vector Integral

Practice Generator

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