# Why resolve some forces into components and not others?

I'm looking at a basic free body diagram of a car travelling on a banked (inclined) road with inclination theta. Its mass is $m$. Here the diagram shows weight ($mg$), normal reaction force ($N$), and centripetal force ($mv^2/r$). They have resolved $N$ as $N\cosθ$ upwards and $N\sinθ$ horizontally: \begin{align} N\cosθ &= mg \\ N\sinθ &= \frac{mv^2}{r} \end{align} My question is, why can't we resolve $mg$ as $mg\cosθ$, and equate it to $N$, normal to the plane? Like $mg\cosθ=N$. Why is this logic wrong?

You can, but the acceleration $v^2/R$ is also diagonal in that coordinate system, so it has a component in the "normal" direction.
You can resolve forces in any direction you wish and then apply Newton's laws to solve the problem, but we resolve the forces in horizontal direction in the case of banking as centripetal Force is horizontal, which makes calculations easier. It is easy to resolve forces in the direction of acceleration, in this case it is $\frac{mv^2}{r}$ in horizontal direction, which would also have to resolved in the coordinate parallel to inclined plane, thus increasing complexity.