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I know how to find the center of a rotation, like where you connect the side and make bisector with the help of your compass. But in a situation like this how do I get the $a_2$? I'm doing an advance study with steering dynamics and lack the value of $a_2$ I know this might be a little silly, I just want to understand like you know when I ride my bike. This is for the Ackermann geometry steering in cars.

Motorcycle

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The center of rotation is where the line of reaction forces intersect. In the diagram above it is clear that knowning $\ell$ and $\delta$ will yield

$$ R_1 = \frac{\ell}{\sin \delta} \, .$$

If you want to find the dynamic center of rotation, given a rigid body and lets say it's center of mass $C$, if you know the position of the center of mass $\vec{r}_C$, the linear velocity of the center of mass $\vec{v}_C$ and the rotational velocity of the rigid body $\vec\omega$ then the instant center of rotation is

$$ \vec{r}_O = \vec{r}_C + \frac{\vec \omega \times \vec{v}_C}{\vec\omega \cdot \vec \omega} $$

where $\cdot$ is the vector dot product, and $\times$ the vector cross product. The above is true for any point on the rigid body, not only $C$.

In 2D the above simplifies to

$$ (x_O,y_O) = (x_C,y_C) + \left(-\frac{\dot{y}_C}{\omega}, -\frac{\dot{x}_C}{\omega} \right) \, .$$

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