Vector-Valued Functions

The central idea

A vector-valued function $\mathbf r:I\subset\mathbb R\to\mathbb R^n$ assigns a vector to each parameter value. Limits, continuity, and derivatives are defined componentwise in finite dimensions. Its image is a parametrized curve, while the vector $\mathbf r(t)$ may represent position. Input dimension one distinguishes it from a vector field on space.

Definitions, hypotheses, and notation

Continuity requires all components to approach their corresponding vector entries; a failure in one component prevents the vector limit. Norm identities such as $\|\mathbf r\|^2=\mathbf r\cdot\mathbf r$ connect scalar and vector calculus. A constant norm constrains motion to a sphere but does not make the vector constant.

Different parametrizations can share the same image yet encode different velocities. For geometric quantities like arc length, regular monotone reparametrizations preserve the result. For physical motion, the time parametrization is essential because it determines velocity and acceleration.

Conceptual meaning

Several coordinate quantities evolve in synchrony under one parameter. Component graphs show how each coordinate changes with time; the combined image shows the trajectory. The parameter interval determines which portion and orientation are included.

A dependable method and decision rule

1. State the common domain of all components.
2. Evaluate or take limits componentwise.
3. Plot key parameter points in output space.
4. Record orientation from increasing parameter and detect repeated values.
5. Differentiate componentwise for velocity or tangent analysis.

Fully worked example

Graphical or geometric meaning

Horizontal position grows uniformly while vertical position follows a concave-down parabola. The plane trajectory is itself parabolic after eliminating time, but the time parametrization retains speed and direction.

Common mistakes and why they fail

Verification and reasonableness checks

• Verify every component is defined on the stated interval.
• Substitute key times into both position and any eliminated relation.
• Check derivative vector units component by component.

Component limits support vector conclusions

Limits, continuity, derivatives, and integrals of $\mathbf r(t)=\langle x(t),y(t),z(t)\rangle$ are computed componentwise. A vector limit exists exactly when every component limit exists. Continuity therefore inherits the common domain of all components. Differentiation produces a tangent or velocity vector, while integration produces a vector of component antiderivatives plus a constant vector. Norm identities supply checks: if $\|\mathbf r(t)\|$ is constant, then $\mathbf r\cdot\mathbf r'=0$. Domain restrictions in a single component restrict the entire vector function. When interpreting a formula, distinguish the curve traced by position vectors from the collection of velocity vectors; they occupy the same coordinate space but represent different geometric objects and physical units.

Practice

1. Find $\lim_{t\to0}\langle\sin t/t,e^t\rangle$.
2. Differentiate $\langle t^2,1/t\rangle$ where defined.
3. What is the input dimension of a vector-valued curve?

Connections and next steps

• Derivatives of Vector Functions
• Space Curves
• Vector Fields