Radial direction is one dimensional or two? When students were asked a question, give example for one dimensional physical quantity, a student writes radial acceleration. He argues that in polar coordinates, just with $r$ one can give the direction for radial acceleration. Is this correct?
Whilst in cartesian coordinates, radial acceleration has two dimension clearly.
Please someone guide me in right direction
 A: This strongly depends on the exact formulation of the question. However, any component of a 3-dimensional vector is a 1-dimensional "vector". So if you are the teacher, then your should (a) try to improve the question of your test (b) check the consistency of the students answer. If it is consistent, then the answer is correct.
EDIT:
There is nothing special about the cartesian coordinate system. For some problem it is suitable, for others it it not. Hence, I think it is not worthy to discuss, whether a rotation with constant radius in the $xy$-plane is a 1D or 2D motion. 


*

*In cartesian coordinates we get a 2D-motion, because we get two equations
\begin{align}
x(t) &= R \sin(\omega t)\\
y(t) &= R \cos(\omega t)
\end{align}
If you like you could always add $z(t)=0$ and discuss, whether or not
this is a 3D-motion, where the third variable is always constant.

*In polar coordinates we get a 1D-motion, because we get a single equation
$$\varphi(t) = \omega t$$
Again, we could add $z(t) = 0$ and $\rho(t) = R$ and discuss, whether or not this is a 3D-motion.

*One could even take the point of view of Special Relativity and argue, that this is a 4D-motion.

*Finally, one could take string theory and argue that this is a multidimensional motion, where some dimensions are rolled up.


There is no point in trying to define, which answer is correct and which is wrong. These are different descriptions of the same phenomena. So if your student gave you a consistent explanation, why the rotation is a 1D-motion, the student probably deserves the credit. If no explanation was given, check your question, whether or not it was sufficiently formulated.
A: Both are correct. The answer is dependent on the basis that you use for your vector space. Your student is correct about polar coordinates, as the $\text{span}(r)$ is a $1-$dimensional line when $\theta$ is fixed. However, if you use the standard Cartesian coordinates and write $\mathbf{r}=\mathbf{x}+\mathbf{y}$ then the $\text{span}(\mathbf{r})=\text{span}(\mathbf{x})+\text{span}(\mathbf{y})$ which is the $2-$dimensional plane.
