[Math] Vector Calculus: to get the parametric equations of the tangent line, why can’t you just use the derivative of the position vector

Question

As you can see in the example, the author generates the parametric equations of the tangent line through the use of r(3) and r'(3).

Why couldn't the author just use the derivative of the position vector as highlighted?

$$x=2t$$
$$y=2t-1$$
$$z=-7$$

Answer 1

The derivative $r'(t)$ is positioned with its tail (or starting point) at the origin $(0,0)$. So, you need to translate it to the point in question $(9,6,-21)$.