A particle moves in the x-y plane according to the law x=at, y=at `(1-alpha t)` where a and `alpha` are positive constants and t is time. Find the velocity and acceleration vector. The moment `t_(0)` at which the velocity vector forms angle of `90^(@)` with acceleration vector.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Write the position vector
The position of the particle in the x-y plane is given by:
\[ x = at \]
\[ y = at(1 - \alpha t) \]
Thus, the position vector \( \mathbf{r} \) can be expressed as:
\[ \mathbf{r} = x \hat{i} + y \hat{j} = at \hat{i} + at(1 - \alpha t) \hat{j} \]
...