Speed and velocity
In order to make our language more precise, we shall make one further definition in our use of the words speed and velocity. Ordinarily we think of speed and velocity as being the same, and in ordinary language they are the same. But in physics we have taken advantage of the fact that there are two words and have chosen to use them to distinguish two ideas. We carefully distinguish velocity, which has both magnitude and direction, from speed, which we choose to mean the magnitude of the velocity, but which does not include the direction. We can formulate this more precisely by describing how the $x$-, $y$-, and $z$-coordinates of an object change with time. Suppose, for example, that at a certain instant an object is moving as shown in Fig. 9–1. In a given small interval of time $\Delta t$ it will move a certain distance $\Delta x$ in the $x$-direction, $\Delta y$ in the $y$-direction, and $\Delta z$ in the $z$-direction. The total effect of these three coordinate changes is a displacement $\Delta s$ along the diagonal of a parallelepiped whose sides are $\Delta x$, $\Delta y$, and $\Delta z$. In terms of the velocity, the displacement $\Delta x$ is the $x$-component of the velocity times $\Delta t$, and similarly for $\Delta y$ and $\Delta z$:  \begin{equation} \label{Eq:I:9:4} \Delta x=v_x\,\Delta t,\quad \Delta y=v_y\,\Delta t,\quad \Delta z=v_z\,\Delta t. \end{equation}
 \begin{equation} \begin{aligned} \Delta x&=v_x\,\Delta t,\\[.5ex] \Delta y&=v_y\,\Delta t,\\[.5ex] \Delta z&=v_z\,\Delta t. \end{aligned} \label{Eq:I:9:4} \end{equation}
