Say $f$ is a scalar valued function from $\mathbb{R}^n \to \mathbb{R}$. When I learnt about the gradient $\nabla f(\mathbf{x})$ I always thought of it as a column vector in the same space as $\mathbf{x}$. That way, the dot product $\nabla f \cdot \mathbf{v}$ gives the directional derivative in direction $\mathbf{v}$.
All the definitions I can find of the Jacobian of $\mathbf{y} = \psi(\mathbf{x})$ however define it as:
\begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \cdots & \frac{\partial y_1}{\partial x_n}\\ \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_2}{\partial x_n}\\ \vdots&\vdots \\ \frac{\partial y_m}{\partial x_1} & \frac{\partial y_m}{\partial x_2} & \cdots & \frac{\partial y_m}{\partial x_n}\\ \end{bmatrix}
But this would make $\nabla f$ a row vector, which then means the directional derivative is no longer $\nabla f \cdot \mathbf{v}$.
Which way is correct? What are the consequences if I accidently write the Jacobian the opposite way? I have found some similar questions here but none that answer my question directly. I'm still learning this stuff so please explain in simple terms :)