Vector-parametric description of the tangent plane to a function of two variables

Question

The vector-parametric description of the tangent plane to the graph of a function $f: \mathbb{R}^2 \to \mathbb{R}$ at a point $(a, b, f(a, b))$ is the following:

$$\begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} a\\ b\\ f(a, b)\\ \end{bmatrix} + \lambda \begin{bmatrix} 1\\ 0\\ f_x(a, b)\\ \end{bmatrix} + \mu \begin{bmatrix} 0\\ 1\\ f_y(a, b)\\ \end{bmatrix},$$

where $\lambda, \mu \in \mathbb{R}$. So every vector on the plane is a linear combination of the two direction vectors $\begin{bmatrix} 1\\ 0\\ f_x(a, b)\\ \end{bmatrix}$ and $\begin{bmatrix} 0\\ 1\\ f_y(a, b)\\ \end{bmatrix}$. Regarding the vector $\begin{bmatrix} a\\ b\\ f(a, b)\\ \end{bmatrix}$, what are the two scalars necessary to make $f(a, b)$ a linear combination of $f_x(a, b)$ and $f_y(a, b)$, i.e. what are the two scalars $\alpha$ and $\beta$ such that $f(a, b) = \alpha f_x(a, b) + \beta f_y(a, b)$?

Answer 1

No, every vector on the plane is a linear combination of the two direction vectors plus the vector

$$\begin{bmatrix} a\\ b\\ f(a, b)\\ \end{bmatrix}$$

which is necessary for the plane to touch the point $(a,b,f(a,b))$.

In other words, the plane parallel to the tangent plane and passing through the origin is given by

$$\begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \lambda \begin{bmatrix} 1\\ 0\\ f_x(a, b)\\ \end{bmatrix} + \mu \begin{bmatrix} 0\\ 1\\ f_y(a, b)\\ \end{bmatrix}$$

which is a linear subspace and the first vector can be viewed as a translation of this plane from the origin to the point $(a,b,f(a,b))$ which is an affine subspace.