In rectangular coordinates, the natural unit vectors are \(\{\boldsymbol{\hat x},\boldsymbol{\hat y}\}\), which point in the direction of increasing \(x\) and \(y\), respectively. Similarly, in polar coordinates the natural unit vectors are \(\boldsymbol{\hat r}\), which points in the direction of increasing \(r\), and \(\boldsymbol{\hat\phi}\), which points in the direction of increasing \(\phi\).

The unit tangent vector to a parametric curve is the unit vector tangent to the curve which points in the direction of increasing parameter. The principal unit normal vector to a parametric curve is the unit vector perpendicular to the curve “in the direction of bending”, which is the direction of the derivative of the unit tangent vector.

1. Consider the parametric curve \(\boldsymbol{\vec r} = 3\cos\phi\,\boldsymbol{\hat x} + 3\sin\phi\,\boldsymbol{\hat y}\) with \(\phi\in[0,2\pi]\). Calculate the unit tangent vector \(\boldsymbol{\hat T}\) and the principal unit normal vector \(\boldsymbol{\hat N}\) for this curve in terms of \(\boldsymbol{\hat x}\) and \(\boldsymbol{\hat y}\).
2. Consider a circle of radius \(3\) centered at the origin. Determine the unit tangent vector \(\boldsymbol{\hat T}\) and the principal unit normal vector \(\boldsymbol{\hat N}\) for this curve in terms of \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\).
3. Compare your answers.