(a) unit vector, \(\hat r\)= cos θ\(\hat i\)+ sin θ\(\hat j\) (1)

\(\hat \theta\) = −sin θ\(\hat i\)+ cos θ\(\hat j\) (2)

Multiplying eq. (1) by sin θ and eq. (2) by cos θ,

Adding ⇒ \(\hat r\)sin θ + \(\hat \theta\) cos θ =\(\hat j\)

Now, multiplying eq. (1) by cos θ and eq. (2) by xsin θ

n(\(\hat r\)cos θ − \(\hat \theta\) sin θ ) = \(\hat i\)

**(b)** \(\hat r.\hat \theta= (cos\,\theta\hat i+sin\,\theta\hat j).(-sin\,\theta\hat i+cos\,\theta\hat j)\)

= \(-cos\,\theta.sin\,\theta+sin\,\theta.cos\,\theta=0\)

θ = 90º

**(c)** \(\frac{dr}{dt}=\frac{d}{dt}(cos\,\theta\hat i+sin\,\theta\hat j)\)

= \(\omega(-sin\,\theta\hat i+cos\,\theta\hat j)\) ∵ ( \(\frac{d\theta}{dt}=\omega\))

**(d)** As, r = aθ\(\hat r\)

⇒ [a] = L = [M^{0}L^{1}T^{0}]

**(e)** \(v=\frac{dr}{dt}=\frac{d\theta}{dt} \hat r+\theta\frac{d\hat r}{dt}\)

= \(\frac{d\theta}{dt}\hat r+\theta[(-sin\,\theta\hat i+cos\,\theta\hat j)\frac{d\theta}{dt}]\)

v =\(\frac{d\theta}{dt}\hat r+\theta\hat \theta \omega=\omega\hat r+\omega\theta\hat \theta\)

a = \(\frac{dv}{dt}=\frac{d}{dt}[\omega\hat r+\omega\theta\,\hat\theta]=\frac{d}{dt}[\frac{d\theta}{dt} \hat r+\frac{d\theta}{dt}(\theta\,\hat \theta)]\)

= \(\frac{d^r \theta}{dt^2}\hat r+\omega^2\hat\theta+\frac{d^2 \theta}{dt^2}\times \theta\,\hat \theta+\omega^2\hat\theta+w^2\theta(-\hat r)\)

= \((\frac{d^2\theta}{dt^2}-\omega^2\theta)\,\hat r\) +\((2\omega^2+\frac{d^2\theta}{dt^2}\hat \theta)\)