Question

Let (r, \(\theta\)) denote the polar coordinates of a particle moving in a plane. If \(\hat{r}\) and \(\hat{\theta}\) represent the corresponding unit vectors, then

• A. \(\frac{d\hat{r}}{d\theta}=\hat{\theta}\)
• B. \(\frac{d\hat{r}}{dr}=-\hat{\theta}\)
• C. \(\frac{d\hat{\theta}}{d\theta}=-\hat{r}\)
• D. \(\frac{d\hat{\theta}}{dr}=\hat{r}\)

Answer 1

The Correct Option is A, C

In this problem, we are given a particle moving in a plane with polar coordinates \((r, \theta)\). We need to establish the relationships between the unit vectors \(\hat{r}\) and \(\hat{\theta}\) when differentiated with respect to \(\theta\).

The unit vectors in polar coordinates are given by:

• \(\hat{r}\): the radial direction, pointing outwards from the origin to the point in consideration.
• \(\hat{\theta}\): the tangential direction, orthogonal to \(\hat{r}\), in the direction of increasing \(\theta\).

In polar coordinates, these unit vectors can be expressed in Cartesian coordinates as:

• \(\hat{r} = \cos(\theta) \hat{i} + \sin(\theta) \hat{j}\)
• \(\hat{\theta} = -\sin(\theta) \hat{i} + \cos(\theta) \hat{j}\)

Let's differentiate these unit vectors with respect to \(\theta\):

1. Differentiating \(\hat{r}\) with respect to \(\theta\): 
   \(\frac{d\hat{r}}{d\theta} = \frac{d}{d\theta}[\cos(\theta) \hat{i} + \sin(\theta) \hat{j}] = -\sin(\theta) \hat{i} + \cos(\theta) \hat{j} = \hat{\theta}\)
2. Differentiating \(\hat{\theta}\) with respect to \(\theta\): 
   \(\frac{d\hat{\theta}}{d\theta} = \frac{d}{d\theta}[-\sin(\theta) \hat{i} + \cos(\theta) \hat{j}] = -\cos(\theta) \hat{i} - \sin(\theta) \hat{j} = -\hat{r}\)

Thus, the correct relationships are:

• \(\frac{d\hat{r}}{d\theta} = \hat{\theta}\)
• \(\frac{d\hat{\theta}}{d\theta} = -\hat{r}\)