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Displacement

Displacement is the change in position of an object with respect to a given reference point or observer.
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The statement of the theorem

Let r:[t1,t2]→R3\mathbf{r}: [t_1, t_2] \to \mathbb{R}^3 be a continuous and differentiable position vector function describing the trajectory of a particle in Cartesian coordinates. The displacement vector Δr\Delta \mathbf{r} between time t1t_1 and time t2t_2 is formally defined as the difference between the final position vector r(t2)\mathbf{r}(t_2) and the initial position vector r(t1)\mathbf{r}(t_1): \n\nΔr=r(t2)−r(t1)\Delta \mathbf{r} = \mathbf{r}(t_2) - \mathbf{r}(t_1) \n\nIf r(t)=⟨x(t),y(t),z(t)⟩\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle, then the components of the displacement are given by:\n\nΔr=⟨x(t2)−x(t1),y(t2)−y(t1),z(t2)−z(t1)⟩\Delta \mathbf{r} = \langle x(t_2) - x(t_1), y(t_2) - y(t_1), z(t_2) - z(t_1) \rangle