A particle moves such that its position vector $\vec{r} (t) = \cos ω t \hat{i} + \sin ω t \hat{j}$ where $ω$ is a constant and t is time. Then which of the following statements is true for the velocity $\vec{ν} (t)$ and $\vec{a} (t)$ acceleration of the particle :

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$\vec{ν}$ is perpendicular to $\vec{r}$ and $\vec{a}$ is directed towards the origin

$\vec{ν}$ and $\vec{a}$ both are perpendicular to $\vec{r}$

$\vec{ν}$ and $\vec{a}$ both are parallel to $\vec{r}$

$\vec{ν}$ is perpendicular to $\vec{r}$ and $\vec{a}$ is directed away from the origin

answer is A.

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Detailed Solution

$\begin{array}{l} \vec{r} = \cos ω t \overset{\land}{i} + \sin ω t \overset{\land}{j} \\ \Rightarrow \vec{v} = \frac{d \vec{r}}{d t} = ω (- \sin ω t \overset{\land}{i} + \cos ω t \overset{\land}{j}) \\ \Rightarrow \vec{a} = \frac{d \vec{v}}{d t} = ω^{2} (- \cos ω t \overset{\land}{i} - \sin ω t \overset{\land}{j}) \\ \Rightarrow \vec{a} = - ω^{2} (\cos ω t \overset{\land}{i} + \sin ω t \overset{\land}{j}) \\ \Rightarrow \vec{a} = - ω^{2} \vec{r} \\ ∴ \vec{a} i s a n t i p a r a l l e l t o \vec{r} \\ \vec{r} i s r a d i a l l y o u t w a r d \\ S o, \vec{a} i s r a d i a l l y i n w a r d \\ A l s o ., \vec{v} . \vec{r} = ω (- \sin ω t \cos ω t + \cos ω t \sin ω t) = 0 \\ S o \vec{v} ⊥ \vec{r} \end{array}$