Why does $h(\vec{x} + t\hat{v}) = h(x + t\hat{v}_x, y + t\hat{v}_y)$ in a multivariable context where $\vec{x} = \langle x, y \rangle$ In the context of multivariable calculus with 2 variables, where we take $\vec{x} + t\hat{v}$ to be the position vector of something dependent on time, why is this linear approximation true?
$\displaystyle{h(\vec{x} + t\hat{v}) = h(x + t\hat{v}_x, y + t\hat{v}_y) \approx h(x,y) + \frac{\partial h}{\partial x}(x,y) \cdot t\hat{v}_x + \frac{\partial h}{\partial y}(x,y) \cdot t\hat{v}_y = h(x) + t(h_x\hat{v}_x + h_y\hat{v}_y)}$
Also, do the dots behind $t\hat{v}_x$ and $t\hat{v}_y$ represent multiplication or dot products? If they represent dot products, why?
Edit: I feel my question lacks some clarity, the parts with which I am confused are why $\displaystyle{h(\vec{x} + t\hat{v}) = h(x + t\hat{v}_x, y + t\hat{v}_y)}$ and $\displaystyle{h(x,y) + \frac{\partial h}{\partial x}(x,y) \cdot t\hat{v}_x + \frac{\partial h}{\partial y}(x,y) \cdot t\hat{v}_y = h(x) + t(h_x\hat{v}_x + h_y\hat{v}_y)}$ are true
 A: Regarding the equation $h(\vec{x} + t\hat{v}) = h(x + t\hat{v}_x, y + t\hat{v}_y)$, the idea is that the inputs on each side of the equation are actually equal:
$$\vec x + t \hat v = (x + t \hat v_x, y + t \hat v_y)
$$
and therefore the outputs are equal, because $h$ is a function.
Let's break this down a bit more:
\begin{align*}
\vec x + t \hat v &= (x,y) + t (\hat v_x,\hat v_y) \\
 &= (x,y) + (t \hat v_x, t \hat v_y) \\
&= (x + t \hat v_x, y + t \hat v_y)
\end{align*}
In the first equation, I am just substituting using $\vec x = (x,y)$ and $\hat v = (\hat v_x,\hat v_y)$. In the second equation, I am applying the formula for scalar multiplication of vectors. In the third equation, I am applying the formula for vector addition.
Regarding the equation
$$h(x,y) + \frac{\partial h}{\partial x}(x,y) \cdot t\hat{v}_x + \frac{\partial h}{\partial y}(x,y) \cdot t\hat{v}_y = h(x) + t(h_x\hat{v}_x + h_y\hat{v}_y)
$$
first there's an error, it looks like it should instead be
$$h(x,y) + \frac{\partial h}{\partial x}(x,y) \cdot t\hat{v}_x + \frac{\partial h}{\partial y}(x,y) \cdot t\hat{v}_y = h(x,y) + t(h_x\hat{v}_x + h_y\hat{v}_y)
$$
Other than that, its just a matter of substituting the notations $h_x = \frac{\partial h}{\partial x}(x,y)$ and $h_y = \frac{\partial h}{\partial y}(x,y)$ and then applying the distributive law of scalar multiplication over vector addition.
