Zero Total Force = Straight Line Motion

$y$

$x$

 $$\large \vec{r}=(x_0 +v_{x0}t)\,\hat{x}+(y_0+v_{y0}t)\,\hat{y}+(z_0+v_{z0}t)\,\hat{z}$$ $$\large \vec{v}=v_{x0}\,\hat{x}+v_{y0}\,\hat{y}+v_{z0}\,\hat{z}$$ $$\large \vec{a}=0$$ $$\large \vec{F}=0$$

$x_0=0$ m   $y_0=0$ m

$v_{x0}=$ 0.5 [m/s]

$v_{y0}=$ 0.5 [m/s]

When the total force acting on a particle is zero, $\vec{F}=0$, the acceleration is also zero,

$$\vec{a}=0.$$

The acceleration is the rate of change of the velocity $\vec{a}=\frac{d\vec{v}}{dt}$ so if the accelleration is zero, the velocity does not change and the three components of the velocity vector are constants,

$$ \vec{v}=v_{x0}\,\hat{x}+v_{y0}\,\hat{y}+v_{z0}\,\hat{z}.$$

The position of a particle moving with constant velocity increases linearly with time,

$$ \vec{r}=(v_{x0}t+x_0)\,\hat{x}+(v_{y0}t+y_0)\,\hat{y}+(v_{z0}t+z_0)\,\hat{z}.$$

When the total force on a particle is zero, it moves with a constant velocity in a straight line. Conversely, if a particle moves with a constant velocity in a straight line, the total force on the particle is zero.