Why does the vector in the banked turn problem not point to the center?

Question

I am currently working on the banked turn problem. I know that once you set up all the equations properly you end up with $$v^2=gR\tan{\theta}$$

I drew a little diagramm of the forces:

I know that the right way to do it is to separate the Normal force $N$ into the two lighblue components $\color{blue}{N\sin{\theta}=\frac{mv^2}{R}}$ $\color{blue}{N\cos{\theta}=mg}$ and after dividing the equations I arrive at the relation $v^2=gR \tan{\theta}$

My question is this: Why do we separate into the blue components and not into something containing the red component. In other words, isn't the red component the one pointing in the direction of the centripetal force $\frac{mv^2}{R}$?

Answer 1

The red line isn't really pointing in the direction of the centripetal force.

Consider the top-down view of the situation, as compared to the side-view we have now. To maintain a curvature, the vector has to point inwards, which would correspond to the horizontal blue line.

The red line you call out would be pointing both towards the centre of the circle, and downwards. This downwards component force wouldn't be contributing to the circular motion, therefore the red vector would not be the centripetal force.