Topic 7 / 17intermediate

Calculus of Vector Functions & Arc Length

Differentiate and integrate vector-valued functions component by component. Compute the unit tangent vector, tangent line, position–velocity–speed, and arc length; reparameterize by arc length.

All single-variable calculus operations on $r (t)$ act componentwise. The first derivative $r^{'} (t)$ gives the velocity (tangent direction), the magnitude $∣ r^{'} (t) ∣$ gives the speed, and integrating $∣ r^{'} (t) ∣$ gives arc length.

Derivatives and integrals — componentwise

r^{'} (t) = ⟨ x^{'} (t), y^{'} (t), z^{'} (t)⟩

. Indefinite integral:

\int r (t) d t = ⟨ \int x d t, \int y d t, \int z d t ⟩ + C

Key takeaways

Component rules from single-variable calculus apply unchanged.
Initial conditions fix the integration constant vector $C$ .

Position, velocity, speed

Position

= r (t)

. Velocity

= r^{'} (t)

. Speed

= ∣ r^{'} (t) ∣

. Acceleration

= r^{''} (t)

. Speed is a scalar; velocity is a vector.

Unit tangent vector and tangent line

Unit tangent:

T (t) = \frac{r ^{'} ( t )}{∣ r ^{'} ( t ) ∣}

. Tangent line at

t = t_{0}

L (s) = r (t_{0}) + s r^{'} (t_{0})

(parameter

s

free).

Strategy tips

You don't need to normalize $r^{'} (t_{0})$ to write the tangent line — any nonzero scalar multiple works as direction.

Differentiation rules for vectors

\frac{d}{d t} (u \cdot v) = u^{'} \cdot v + u \cdot v^{'}

and

\frac{d}{d t} (u \times v) = u^{'} \times v + u \times v^{'}

(order matters!). Chain rule for scalar substitution:

\frac{d}{d t} r (g (t)) = g^{'} (t) r^{'} (g (t))

Arc length

Length of

r (t)

from

t = a

t = b

L = \int_{a}^{b} ∣ r^{'} (t) ∣ d t

. The integrand

∣ r^{'} (t) ∣ d t = d s

is the arc length element.

Arc length parameter and reparameterization

Let

s (t) = \int_{t_{0}}^{t} ∣ r^{'} (u) ∣ d u

. A curve is **arc length parameterized** iff

∣ r^{'} (s) ∣ = 1

for all

s

. To reparameterize by arc length: solve

s = s (t)

for

t

in terms of

s

, then substitute into

r (t)

Strategy tips

$∣ r^{'} (t) ∣ = 1$ at all $t$ is the test for arc length parameterization.
If $∣ r^{'} (t) ∣$ is constant but not 1, scale to make it 1.

Worked examples

Example 1

For

r (t) = ⟨ t^{3}, t^{2}, t ⟩

, find

r^{'} (t)

and the speed at

t = 1

1
Differentiate componentwise.
$r^{'} (t) = ⟨ 3 t^{2}, 2 t, 1 ⟩$
2
Plug in $t = 1$ .
$r^{'} (1) = ⟨ 3, 2, 1 ⟩$
3
Speed is the magnitude.
$∣ r^{'} (1) ∣ = 9 + 4 + 1 = 14$

Answer

r^{'} (t) = ⟨ 3 t^{2}, 2 t, 1 ⟩

; speed at

t = 1

14

Example 2

Find the tangent line to

r (t) = ⟨ 2 sin t, 2 sin 2 t, 2 sin 3 t ⟩

at the point

(1, 3, 2)

1
Find $t$ . Try $t = π /6$ : $2 sin (π /6) = 1$ , $2 sin (π /3) = 3$ , $2 sin (π /2) = 2$ . ✓
2
Compute $r^{'} (t)$ .
$r^{'} (t) = ⟨ 2 cos t, 4 cos 2 t, 6 cos 3 t ⟩$
3
Evaluate at $t = π /6$ .
$r^{'} (π /6) = ⟨ 3, 2, 0 ⟩$
4
Tangent line passes through $(1, 3, 2)$ with direction $⟨ 3, 2, 0 ⟩$ .
$L (s) = (1, 3, 2) + s ⟨ 3, 2, 0 ⟩$

Answer

L (s) = (1 + s 3, 3 + 2 s, 2)

Example 3

Find the arc length of

r (t) = ⟨ t, t^{2}, \frac{2 t ^{3/2}}{3} \cdot 2 ⟩

from

t = 0

t = 1

. (Use

r (t) = ⟨ t^{2}, \frac{4 2}{3} t^{3/2}, 2 t ⟩

instead.)

1
Compute $r^{'} (t)$ for $r (t) = ⟨ t^{2}, \frac{4 2}{3} t^{3/2}, 2 t ⟩$ .
$r^{'} (t) = ⟨ 2 t, 22 t^{1/2}, 2 ⟩$
2
Magnitude squared.
$∣ r^{'} (t) ∣^{2} = 4 t^{2} + 8 t + 4 = 4 (t + 1)^{2}$
3
Magnitude.
$∣ r^{'} (t) ∣ = 2 (t + 1)$
4
Integrate from 1 to 3 (Final Review #40).
$L = \int_{1}^{3} 2 (t + 1) d t = [(t + 1)^{2}]_{1}^{3} = 16 - 4 = 12$

Answer

L = 12

Interactive visualizations

Loading visualization…

Helix with the unit tangent vector drawn at the current

t

. The orange arrow shows the direction of motion.

Formulas in this topic

Derivative

r^{'} (t) = ⟨ x^{'} (t), y^{'} (t), z^{'} (t)⟩

Componentwise differentiation.

Unit tangent

T (t) = r^{'} (t) /∣ r^{'} (t) ∣

Direction of motion, normalized.

Arc length

L = \int_{a}^{b} ∣ r^{'} (t) ∣ d t

Integral of speed.

Arc length element

d s = ∣ r^{'} (t) ∣ d t

Differential of arc length.

Tangent line

L (s) = r (t_{0}) + s r^{'} (t_{0})

Line tangent to curve at parameter $t_0$.

Derivatives and integrals — componentwise

Position, velocity, speed

Unit tangent vector and tangent line

Differentiation rules for vectors

Arc length

Arc length parameter and reparameterization

Worked examples

Interactive visualizations

Formulas in this topic

Related topics