A moving point

Now consider the motion of a point in a 3D world ${\mathbb{R}}^3$. Imagine that a geometric model, as defined in Section 3.1, is moving over time. This causes each point $(x,y,z)$ on the model to move, resulting a function of time for each coordinate of each point:

$$(x(t),y(t),z(t)) .$$ (8.8)

The velocity $v$ and acceleration $a$ from Section 8.1.1 must therefore expand to have three coordinates. The velocity $v$ is replaced by $(v_x,v_y,v_z)$ to indicate velocity with respect to the $x$, $y$, and $z$ coordinates, respectively. The magnitude of $v$ is called the speed:

$$\sqrt{v_x^2 + v_y^2 + v_z^2}$$ (8.9)

Continuing further, the acceleration also expands to include three components: $(a_x,a_y,a_z)$.