Question

Relation between linear acceleration and angular acceleration in vector form

Answer 1

There could be 2 interpretations of your question -

Relationship between angular acceleration and linear acceleration (as in linear motion or translation motion)

In linear motion if a particle moves from a velocity Vi at time Ti to Vf at time Tf, we say acceleration of the particle is change in velocity (Vf - Vi) divided by time taken (Tf- Ti). This is average acceleration, instantaneous would be dv/dt

Angular acceleration is commonly * used for describing motion of a body in rotational motion. Here, angular acceleration is the rate of change of angular velocity between time Tf and Ti given by (Wf - Wi)/ (Tf - Ti). Instantaneous angular acceleration would be dw/dt

Thus you see that this is an analogy and there is no connection between the two. Watch this video for a better understanding

Rotation - Angular Velocity | Displacement | Acceleration #1

2. The two entities considered within rotational motion

Consider a particle that has an angular velocity w and angular acceleration α.

In such a case the tangential acceleration would be Ta = rα and the centripetal acceleration would be Ca = rω(sq) (r being the position vector from axis of the point). The linear acceleration would then be the vector sum of the two

*a particle moving in a straight line can also have angular velocity/ acceleration about an axis