Firstly, the matrix notation of the series is,
\begin{align*}
f\left(x, y, z\right) &= f\left(a, b, c\right) + \left(\mathbf{x} - \mathbf{a}\right)^T \frac{\partial f\left(a, b, c\right)}{\partial \mathbf{x}} + \frac{1}{2}\left(\mathbf{x} - \mathbf{a}\right)^T \frac{\partial^2 f\left(a, b...