Topic 6 / 17intermediate

Vector-Valued Functions

Parameterize curves in 2D and 3D, find domains and limits of vector functions, eliminate parameters, and use coordinate-plane projections to visualize 3D curves.

A vector-valued function $r (t) = ⟨ x (t), y (t), z (t)⟩$ traces a curve as $t$ varies. The output of $r$ is a vector (or, equivalently, a point); the input $t$ is a single real parameter — typically time, angle, or arc length. Many concepts from single-variable calculus extend to vector functions component by component.

Domain of a vector function

The domain of

r (t) = ⟨ x (t), y (t), z (t)⟩

is the intersection of the domains of its components. Each component must be defined: check denominators, square roots, logs.

Limits of vector functions

lim_{t \to a} r (t) = ⟨ lim_{t \to a} x (t), lim_{t \to a} y (t), lim_{t \to a} z (t) ⟩

. Compute componentwise. If any component limit fails, the whole limit fails.

Standard parameterizations

Common curves you should be able to parameterize from memory: - Line segment from

A

B

r (t) = (1 - t) A + t B

0 \leq t \leq 1

. - Circle of radius

a

, centered at origin:

⟨ a cos t, a sin t ⟩

0 \leq t \leq 2 π

. - Circle radius

a

centered at

(h, k)

⟨ h + a cos t, k + a sin t ⟩

. - Ellipse

x^{2} / a^{2} + y^{2} / b^{2} = 1

⟨ a cos t, b sin t ⟩

. - Helix along the

z

-axis:

⟨ cos t, sin t, t ⟩

. - Parabola

y = x^{2}

⟨ t, t^{2} ⟩

Strategy tips

If asked to parameterize $y = f (x)$ , set $x = t$ and $y = f (t)$ .
For an ellipse or circle, $cos$ and $sin$ scaled by axis lengths usually works.

Eliminate parameter to sketch

Solve one equation for

t

and substitute into the others to find a Cartesian relation. Example:

x = cos t, y = sin t \Rightarrow x^{2} + y^{2} = 1

— a unit circle.

3D curves via projections

To understand a curve in 3D, study its projections onto the three coordinate planes. The projection onto the

x y

-plane is

⟨ x (t), y (t), 0 ⟩

; into

x z

⟨ x (t), 0, z (t)⟩

; into

y z

⟨ 0, y (t), z (t)⟩

. The shape of each projection plus how

z

changes with

t

usually tells you the whole story.

Worked examples

Example 1

Find the domain of

r (t) = ⟨ \frac{t}{t + 1}, 4 - t^{2}, e^{- t} ⟩

1
First component: $t + 1 \neq = 0 \Rightarrow t \neq = - 1$ .
2
Second: $4 - t^{2} \geq 0 \Rightarrow - 2 \leq t \leq 2$ .
3
Third: defined for all real $t$ .
4
Intersect: $[- 2, - 1) \cup (- 1, 2]$ .

Answer

[- 2, - 1) \cup (- 1, 2]

Example 2

Parameterize the line segment from

A (1, 2, 3)

B (4, 0, - 1)

1
Use the linear interpolation form.
$r (t) = (1 - t) ⟨ 1, 2, 3 ⟩ + t ⟨ 4, 0, - 1 ⟩, 0 \leq t \leq 1$
2
Expand componentwise.
$= ⟨ 1 + 3 t, 2 - 2 t, 3 - 4 t ⟩, 0 \leq t \leq 1$

Answer

r (t) = ⟨ 1 + 3 t, 2 - 2 t, 3 - 4 t ⟩, 0 \leq t \leq 1

Example 3

Eliminate the parameter for

r (t) = ⟨ 3 cos t, 3 sin t, 2 t ⟩

to describe the curve.

1
From the first two components, $x^{2} + y^{2} = 9 cos^{2} t + 9 sin^{2} t = 9$ — a circle of radius 3 in the $x y$ -plane projection.
2
$z = 2 t$ rises linearly with $t$ , so as $t$ increases the curve spirals upward.
3
Combined: a **helix** of radius 3 about the $z$ -axis.

Answer

A helix of radius 3 about the

z

-axis, with

z = 2 t

(rising 4π per turn).

Interactive visualizations

Loading visualization…

Helix

r (t) = ⟨ 3 cos t, 3 sin t, t /2 ⟩

. Dashed lines show projections into

x y

and

x z

planes.

Formulas in this topic

Line segment

r (t) = (1 - t) A + t B, 0 \leq t \leq 1

Parametric form for a segment from A to B.

Circle of radius a at (h,k)

r (t) = ⟨ h + a cos t, k + a sin t ⟩

Standard circle parametrization.

Ellipse with semi-axes a, b

r (t) = ⟨ a cos t, b sin t ⟩

Standard ellipse parametrization.

Helix

r (t) = ⟨ a cos t, a sin t, c t ⟩

Spiraling curve about z-axis.

Domain of a vector function

Limits of vector functions

Standard parameterizations

Eliminate parameter to sketch

3D curves via projections

Worked examples

Interactive visualizations

Formulas in this topic

Related topics