Derivative of Vector Functions

Definition

The derivative of a vector-valued function $\mathbf{r}(t)$ , denoted $\mathbf{r}'(t)$ or $d\mathbf{r}/dt$ , represents the instantaneous rate of change of the vector with respect to its parameter. Geometrically, it is a vector tangent to the curve traced by $\mathbf{r}(t)$ at any given point.

Why It Matters

Vector derivatives are the language of motion in 3D space. They are indispensable for aerospace and robotics, providing the exact velocity and acceleration vectors needed to guide vehicles and arms along complex, curved paths.

Core Concepts

Component-wise Differentiation: The derivative is found by differentiating each component function independently: $\mathbf{r}'(t) = f'(t)\mathbf{i} + g'(t)\mathbf{j} + h'(t)\mathbf{k}$ $r^{'} (t) = f^{'} (t) i + g^{'} (t) j + h^{'} (t) k$ .
- How to read: “The derivative r prime of t equals f prime of t times unit vector i, plus g prime of t times unit vector j, plus h prime of t times unit vector k.”
- Meaning / when to use: Differentiate each component separately—vector derivative is component-wise.
Kinematic Interpretations:
- Velocity: $\mathbf{v}(t) = \mathbf{r}'(t)$ $v (t) = r^{'} (t)$ (always tangent to the path).
  - How to read: “The velocity v of t equals the derivative r prime of t.”
  - Meaning: Velocity vector is the derivative of position.
- Speed: $v(t) = |\mathbf{v}(t)|$ $v (t) = ∣ v (t) ∣$ (the magnitude of the velocity vector).
  - How to read: “The speed equals the magnitude of the velocity vector v of t.”
  - Meaning: Scalar speed—how fast, ignoring direction.
- Acceleration: $\mathbf{a}(t) = \mathbf{v}'(t) = \mathbf{r}''(t)$ $a (t) = v^{'} (t) = r^{''} (t)$ .
  - How to read: “The acceleration a of t equals the derivative v prime of t, which equals the second derivative r double prime of t.”
  - Meaning: Rate of change of velocity—second derivative of position.
Constant Length Property: If a vector function has a constant magnitude ( $|\mathbf{r}(t)| = c$ $∣ r (t) ∣ = c$ ), then the vector and its derivative are always orthogonal: $\mathbf{r} \cdot \mathbf{r}' = 0$ $r \cdot r^{'} = 0$ .
- How to read: “The dot product of r and r prime equals zero.”
- Meaning: For motion on a sphere/circle, velocity is perpendicular to position.

Derivative of Vector Functions

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes