Motion in three dimensions

Position in three dimensions is a vector whose tail is the reference point and whose tip is the object. It can be given by the \(x, y, z\) coordinates:

\[\mathbf{r} = (x, y, z).\]

Displacement is, by definition, the change in the position vector:

\[\Delta\mathbf{r} = \mathbf{r} - \mathbf{r}_0 = \left(x, y, z\right) - \left(x_0, y_0, z_0\right) = \left(x - x_0, y - y_0, z - z_0\right) = \left(\Delta x, \Delta y, \Delta z\right).\]

Average velocity is obtained dividing the displacement by the time interval \(\Delta t\) then multiplying by the scalar \(1/\Delta t\):

\[\overline{\mathbf{v}} = \frac{\Delta\mathbf{r}}{\Delta t} = \frac{1}{\Delta t} \Delta\mathbf{r} = \frac{1}{\Delta t} \left(\Delta x, \Delta y, \Delta z\right) = \left(\frac{\Delta x}{\Delta t}, \frac{\Delta y}{\Delta t}, \frac{\Delta z}{\Delta t}\right) = \left(\overline{v_x}, \overline{v_y}, \overline{v_z}\right).\]

Velocity is the limit of the average velocity as \(\Delta t\) approaches zero. This limit must be calculated component by component:\[\mathbf{v} = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} = \lim\limits_{\Delta t \to 0} \frac{\Delta\mathbf{r}}{\Delta t} = \lim\limits_{\Delta t \to 0}\left(\frac{\Delta x}{\Delta t}, \frac{\Delta y}{\Delta t}, \frac{\Delta z}{\Delta t}\right) = \left(\lim\limits_{\Delta t \to 0}\frac{\Delta x}{\Delta t}, \lim\limits_{\Delta t \to 0}\frac{\Delta y}{\Delta t}, \lim\limits_{\Delta t \to 0}\frac{\Delta z}{\Delta t}\right)\]

\[\mathbf{v} = \left(\frac{\mathrm{d}x}{\mathrm{d}t}, \frac{\mathrm{d}y}{\mathrm{d}t}, \frac{\mathrm{d}z}{\mathrm{d}t}\right) = \left(v_x, v_y, v_z\right)\]

Acceleration is defined as the first derivative of velocity with respect to time; again, this derivative must be calculated component by component:\[\mathbf{a} = \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = \lim\limits_{\Delta t \to 0} \frac{1}{\Delta t}\left(\Delta v_x, \Delta v_y, \Delta v_z\right) = \left(\lim\limits_{\Delta t \to 0}\frac{\Delta v_x}{\Delta t}, \lim\limits_{\Delta t \to 0}\frac{\Delta v_y}{\Delta t}, \lim\limits_{\Delta t \to 0}\frac{\Delta v_z}{\Delta t}\right)\]\[\mathbf{a} = \left(\frac{\mathrm{d}v_x}{\mathrm{d}t}, \frac{\mathrm{d}v_y}{\mathrm{d}t}, \frac{\mathrm{d}v_z}{\mathrm{d}t}\right) = \left(a_x, a_y, a_z\right) = \left(\frac{\mathrm{d}^2x}{\mathrm{d}t^2}, \frac{\mathrm{d}^2y}{\mathrm{d}t^2}, \frac{\mathrm{d}^2z}{\mathrm{d}t^2}\right)\]

This shows us that three-dimensional motion can be broken down into the combination of three one-dimensional motion components.