Vector-Valued Functions and Motion

Vector-Valued Functions

By the end of this lesson, you will be able to differentiate and integrate vector-valued functions, calculate position, velocity, speed, and acceleration from a vector function, and find arc length.

Vector-Valued Functions

A vector-valued function in the plane assigns a vector (not a scalar) to each value of t:

$$\vec{r}(t) = \langle x(t), y(t) \rangle = x(t)\hat{i} + y(t)\hat{j}$$

This traces a curve in the plane as t varies.

Component functions: x(t) and y(t) are standard real-valued functions.

Parametric Equations Connection

A vector-valued function is equivalent to a pair of parametric equations: $$x = x(t), \quad y = y(t)$$

Example: r(t) = ⟨cos t, sin t⟩ traces the unit circle as t goes from 0 to 2π.

Derivatives of Vector-Valued Functions

Differentiate component by component:

$$\vec{r}'(t) = \langle x'(t), y'(t) \rangle$$

Interpretation:

• $\vec{r}'(t)$ = velocity vector (when t is time) — points in the direction of motion
• Direction of $\vec{r}'(t)$ = direction of the tangent line

Higher Derivatives

$$\vec{r}''(t) = \langle x''(t), y''(t) \rangle = \text{acceleration vector}$$

Speed and Arc Length

Speed (scalar, always non-negative):

$$|\vec{v}(t)| = |\vec{r}'(t)| = \sqrt{[x'(t)]^2 + [y'(t)]^2}$$

Arc length from t = a to t = b:

$$L = \int_a^b |\vec{r}'(t)| , dt = \int_a^b \sqrt{[x'(t)]^2 + [y'(t)]^2} , dt$$

Total distance traveled (even when the curve doubles back):

$$\text{Distance} = \int_a^b |\vec{v}(t)| , dt$$

This is the same formula — arc length is total distance traveled.

Displacement vector (net change in position):

$$\Delta\vec{r} = \vec{r}(b) - \vec{r}(a) = \int_a^b \vec{r}'(t) , dt$$

Integration of Vector-Valued Functions

$$\int \vec{r}(t) , dt = \left\langle \int x(t) , dt, \int y(t) , dt \right\rangle + \vec{C}$$

Where $\vec{C} = \langle C_1, C_2 \rangle$ is a constant vector of integration.