In the Chapter 12 Appendix of Spivak's Calculus (4th Ed), the following paragraph is written about an arbitrary vector valued function $c$, which should be interpreted as $c: \mathbb R \to \mathbb R^2$. (Note that this is the only time in the book thus far that we have dealt with functions outside of $f:\mathbb R \to \mathbb R$)
Limits lead us of course to derivatives. For \begin{align} c(t)=(u(t),v(t))\end{align} we can define $c'$ by the straightforward definition \begin{align} c'(a)=(u'(a),v'(a)). \end{align} We could also try to imitate the basis definition: \begin{align}c'(a)=\displaystyle \lim_{h \to 0}\frac{c(a+h)-c(a)}{h}, \end{align}where the fraction on the right-hand side is understood to mean $\frac{1}{h}\cdot [c(a+h)-c(a)]$. As a matter of fact, these two definitions are equivalent, becaise \begin{align} \displaystyle\lim_{h \to 0}\frac{c(a+h)-c(a)}{h}&=\lim_{h \to 0}\left(\frac{u(a+h)-u(a)}{h},\frac{v(a+h)-v(a)}{h}\right)\\&=\lim_{h \to 0}\left(\lim_{h \to 0}\frac{u(a+h)-u(a)}{h},\lim_{h \to 0}\frac{v(a+h)-v(a)}{h}\right)\\&=\left(u'(a),v'(a)\right)\end{align} Figure $5$ shows $c(a+h)$ and $c(a)$, as well as the arrow from the $c(a)$ to $c(a+h)$; as we showed in Appendix $1$ to Chapter $4$, this arrow is $c(a+h)-c(a)$, except moved over so that it starts at $c(a)$. As $h \to 0$, this arrow would appear to move closer and closer to the tangent of our curve, so it seems reasonable to define $\color{red}{\text{the tangent line of } c \text{ at }c(a) \text{ to be the straight line along }c'(a)}$, when $c'(a)$ is moved over so that it starts at $c(a)$.
Using the attached image ('Figure $5$') as reference, I am hoping that someone could explain to me what Spivak is trying to convey in the sentence I have emphasized in $\color{red}{\text{red}}$.
Firstly, what does he mean by: "...the tangent line of $c$ at $c(a)$..."? Shouldn't this, instead, read as: "...the tangent line of $c$ at $\left(a,c(a) \right)$ "? I ask because when tangent lines are first introduced in the context of Spivak's Calculus, we are discussing functions of $f: \mathbb R \to \mathbb R$ and a tangent line is defined as:
...we define the tangent line to the graph of $f$ a $(a,f(a))$ to be the line through $(a,f(a))$ with slope $f'(a)$.
Secondly, what does he mean by:"...straight line along $c'(a)$..."? Clearly, $c'(a)$ is, itself, a vector...i.e. $c'(a) \in \mathbb R^2$. So I guess he is saying: "Place the line in the direction that $c'(a)$ is pointing"...but, once again, shouldn't the line be oriented in a 3D space (i.e. oriented along the same direction as $(1,c'(a))$?