Can potential energy be treated as vector quantity? Force can be representd by potential energy function which for 1 dimension case satisfies the derivatives condition $f(x)=-d U(x)/d x$.
If we now consider 3D space, then position will now depends on coordinate system $(x,y,z)$. Then force can be expressed in components as:
$$F_x =-\frac{d U(x,y,z)}{dx}.$$
Similarly, $F_y = -dU(x,y,z)/d y$.
Since potential energy is function of position, can we say that along different axis, potential energy is different and hence can be considered as vector quantity? If not how to find potential energy in 3D space?
 A: 
Since potential energy is function of position, ... hence can be considered as vector quantity?

You make a conceptual mistake:
"$n$-dimensional vector quantity" does not mean that a quantity depends on the position in an $n$-dimensional coordinate system.
"$n$-dimensional vector quantity" means that a quantity requires $n$ different real numbers to be described completely.
One example would be the force: You will need 3 different real values ($F_x$, $F_y$ and $F_z$) to describe the force in magnitude and direction.
In engineering (for example in materials science) you will even find 6-dimensional quantities. It is obvious that this does not mean that these quantities depend on 6 axes of the (only) 3-dimensional coordinate system. It means that you require 6 real numbers to correctly describe such a quantity.
And there is a mathematical argument why the potential energy is not a vector:

$F_x = -\frac{dU(x,y,z)}{dx}$, $F_y = -\frac{dU(x,y,z)}{dy}$ ...

If the force $\overrightarrow F$ is constant and we integrate these equations, we get:
$F_x\Delta x + F_y\Delta y + F_z\Delta z = -\Delta U(x,y,z)$
This equation can be written using the dot product notation:
$\overrightarrow{F}\cdot\Delta\overrightarrow p = -\Delta U(\overrightarrow p)$
(While $\overrightarrow p$ is the position.)
However, the result of a dot product of two vectors is a scalar and not a vector.
A: Definition: The change in potential energy of the system is defined as the negative of work done by the internal conservative forces of the system.
Potential energy may vary with space just like mass of a non uniform rod which may be represented as $f(x,y,z)$.
After all potential energy is basically negative of work done by internal conservative forces which is a scalar product.
A: Potential energy is still a scalar quantity even in more than one dimension. This is because it only has a magnitude, there is no direction of potential energy. You can think of it as similar to the temperature in a room. Even though it varies with position, it still does not have a direction.
A: No, just because a value changes over space doesn't mean it is a vector quantity. Potential energy is not a vector. It is a scalar quantity related to its corresponding conservative force by
$$\mathbf F=-\nabla U=-\left(\frac{\partial U}{\partial x}\hat x+\frac{\partial U}{\partial y}\hat y+\frac{\partial U}{\partial z}\hat z\right)$$
So the force components are
$$F_x=-\frac{\partial U}{\partial x}$$
$$F_y=-\frac{\partial U}{\partial y}$$
$$F_z=-\frac{\partial U}{\partial z}$$
