An intresting situation regarding coordinate system. In elementry school we are taught how to represent a point $(x,y)$ in a coordinate system having two mutually perpendicular axis.
At higher level we know that these axis are actually two dimentions of vector space.
Q:Is it necessary that two axis should be mutually perpendicular?
we know our x-axis and y-axis are along direction$(1,0)$ and $ (0,1)$
Q: Cant we change to non arthogonal directions $(1,2) , (1,3)$ ??
Now the inclinattion is $cos^{-1}\frac{6}{5\sqrt12}$
Q:Under new coordinate system how the scaling of axes is affected?
I mean if previposly in orthogonal coordinate the unit of measurement  was 1 cm them how new axes are to be marked?
While representing a point (1,3) in orthogonal coordinate system we move 1 unit in the direction of x-axis and 3 units in y direction.How can we represent such a point on axes which are inclined at sixty degree?
 A: Suppose that $(1,3)=\alpha(1,0)+\beta(\frac{1}{2},\frac{1}{\sqrt{3}})$. We clearly need $\beta=3\sqrt{3}$. Then $\alpha+3\sqrt{3}\frac{1}{2}=1$ implies that $\alpha=1-3\sqrt{3}\frac{1}{2}$.
The point $(1,3)$ is then given by $(1-\frac{3\sqrt{3}}{2})(1,0)+3\sqrt{3}(\frac{1}{2},\frac{1}{\sqrt{3}}),$
where the vectors $(1,0)$ and $(\frac{1}{2},\frac{1}{\sqrt{3}})$ are vectors which make an angle of $60$ degrees.
A: Q1)
Ans) No
Q2)
Ans) Yes, you can. This is because of the factthat any two vector spaces of same dimention are isomorphic.Changing basis rotates your points.
Q3)
Ans)
The isomorpism from vector space $V$ of dimention $n$ to $V'$ of same dimention over same field $F$ with basis $B=\{x_1,...,x_n\}$ is given by $f(\alpha)=a_1x_1+...+a_nx_n, a_i \in F$
Now you dnt need to worry about new axis.
Consider the vector space with basis $\{(1,0), (1,\sqrt3)\}$ 
You can varify new axis are inclined at sixty degree. Suppose you want to represent $(1,1)$ on this new system.
Write $$(1,1)=\frac{\sqrt3-1}{\sqrt3}(1,0)+\frac{1}{\sqrt3}(1,\sqrt3)$$
your coordinate vector of $(1,1)$ is  $$(\frac{\sqrt3-1}{\sqrt3},\frac{1}{\sqrt3})$$
Represent your coordinate vector in $\{(1,0),(0,1)\}$. This will correspond to $(1,1)$ in $\{(1,0),(1,\sqrt3)\}$..
