Position Vectors
In 3D space, the position of a particle is a vector \(\vec{r}(t) = \langle x(t), y(t), z(t) \rangle\). Its movement is described by the derivatives of this vector:
- Velocity: \(\vec{v}(t) = \frac{d\vec{r}}{dt} = \langle x', y', z' \rangle\).
- Acceleration: \(\vec{a}(t) = \frac{d\vec{v}}{dt} = \langle x'', y'', z'' \rangle\).
Worked Examples
Example 1: Circular Motion
A particle moves along \(\vec{r}(t) = \langle \cos t, \sin t, 0 \rangle\).
- \(\vec{v}(t) = \langle -\sin t, \cos t, 0 \rangle\).
- \(\vec{a}(t) = \langle -\cos t, -\sin t, 0 \rangle\).
- Notice that \(\vec{a}(t) = -\vec{r}(t)\). The acceleration points directly toward the center of the circle. This is Centripetal Acceleration.
The Bridge to Quantum Mechanics
In Quantum Mechanics, we don't track the position vector \(\vec{r}(t)\) of a particle. Instead, we track the "Expected Position" \(\langle \vec{r} \rangle\). According to Ehrenfest's Theorem, the derivative of the expected position is the expected momentum: \(\frac{d}{dt}\langle \vec{r} \rangle = \frac{\langle \vec{p} \rangle}{m}\). This ensures that while individual measurements are probabilistic, the average behavior of a quantum particle perfectly matches the vector calculus of classical mechanics.