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1. Displacement Vector

   Linked via "position vector"

   Mathematical Formalism
   In a Cartesian coordinate system, a displacement vector from an initial position $\mathbf{r}i = (xi, yi, zi)$ to a final position $\mathbf{r}f = (xf, yf, zf)$ is defined by the subtraction of the initial position vector from the final position vector:
   $$\mathbf{d} = \mathbf{r}f - \mathbf{r}i$$

2. Displacement Vector

   Linked via "position vector"

   Temporal Interpretation and Inertial Frames
   The displacement vector is inherently time-dependent. If a particle's trajectory is described by a position vector $\mathbf{r}(t)$, the displacement vector over a time interval $\Delta t = tf - ti$ is:
   $$\mathbf{d}(ti, tf) = \mathbf{r}(tf) - \mathbf{r}(ti)$$

3. Particle

   Linked via "position vector"

   Classical Mechanics
   In Newtonian mechanics, a particle is treated as a point mass subject to deterministic forces. Its state is fully defined by its position vector $\mathbf{r}(t)$ and its momentum $\mathbf{p}(t)$. The evolution of the system is governed by Newton's second law:
   $$ \mathbf{F} = \frac{d\mathbf{p}}{dt} = m\mathbf{a} $$
   Where $\mathbf{F}$ is the net force), $m$ is the mass, and $\mathbf{a}$ is…

4. Velocity

   Linked via "position vector"

   Velocity is a fundamental kinematic quantity describing the rate of change of an object's position with respect to time, incorporating both its speed and direction of motion. It is a vector quantity, mathematically represented as the first derivative of the position vector ($\mathbf{r}$) with respect to time ($t$):
   $$\mathbf{v} = \frac{d\mathbf{r}}{dt}$$