Newton's laws connect interactions, represented by forces, with changes in motion.

First law: inertia

If the net external force on an object is zero, its velocity remains constant:

$$\mathbf F_{\mathrm{net}}=\mathbf0 \quad\Longrightarrow\quad \mathbf v=\text{constant}.$$

Rest is only the special case of constant velocity with $\mathbf v=0$.

Second law: force and acceleration

For an object of constant mass $m$,

$$\sum\mathbf F=m\mathbf a.$$

The acceleration points in the direction of the net force. Mass measures inertia: for the same net force, a larger mass produces a smaller acceleration.

In components,

$$\sum F_x=ma_x,\qquad \sum F_y=ma_y.$$

Free-body diagrams identify the forces that belong in these sums.

Third law: interaction pairs

If object A exerts a force on object B, then B exerts an equal and opposite force on A:

$$\mathbf F_{A\to B}=-\mathbf F_{B\to A}.$$

The two forces act on different objects, so they do not cancel on one object's free-body diagram.

Newton's laws turn a physical model of interactions into equations of motion, but only after the relevant system and forces have been identified.