A particle moves such that its position vector $$\vec { r } (t)=\cos { \omega t } \hat { i } +\sin { \omega t } \hat { j } $$ where $$\omega$$ is a constant and $$t$$ is time. Then which of the following statements is true for the velocity $$\vec { v } (t)$$ and acceleration $$\vec { a } (t)$$ of the particle:
A
$$\vec { v } $$ and $$\vec { a } $$ both are perpendicular to $$\vec { r } $$
B
$$\vec { v } $$ is perpendicular to $$\vec { r } $$ and $$\vec { a } $$ is directed away from the origin
C
$$\vec { v } $$ and $$\vec { a } $$ both are parallel to $$\vec { r } $$
D
$$\vec { v } $$ is perpendicular to $$\vec { r } $$ and $$\vec { a } $$ is directed towards the origin
Correct option is A. $$\vec { v } $$ is perpendicular to $$\vec { r } $$ and $$\vec { a } $$ is directed away from the origin
$$\vec{r}( t) = \cos wt \hat{i} + \sin w t \hat{j}$$$$\vec{v}(t)= \dfrac{d\vec{r}(t)}{dt} =w (-\sin wt \hat{i} + \cos wt\hat{j})$$
$$\vec{a}(t) = \dfrac{d\vec{v}(t)}{dt} = - w^2 (\cos wt\hat{i} + \sin wt{j})$$
$$\vec{a}(t) = -w^2\vec{r}(t)$$, negative sign shows that acceleration is antiparallel to position of positicle
$$\vec{v}(t).\vec{r}(t) = w(-\sin wt \cos wt + \cos wt \sin wt) = 0$$
So, $$\vec{v}(t) $$ is perpendicular to $$\vec{r}(t)$$