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The motion of a particle is defined by the position vector
\( \mathrm{P} \)
\[
\vec{r}=A(\cos t+\mathrm{t} \sin t) \hat{i}+A(\sin t-t \cos t) \hat{j}
\]
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where \( t \) is expressed in seconds. Determine the value of \( t \) for which:

The positions vectors and acceleration vectors are perpendicular
(1) at \( t=1 \mathrm{sec} \)
(2) at \( t=0 \)
(3) at \( t=\sqrt{2} \mathrm{sec} \)
(4) at \( t=1.5 \mathrm{sec} \)
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