Angle vector in polar system represented by Cartesian vector

Question

$x=r\cos\theta,\,y=r\sin\theta\implies r^2=x^2+y^2,\,\theta=\arctan(y/x)$

I can show that $\hat{r}=\cos\theta\hat i+\sin\theta\hat j$, where the hat vectors are unit, and $\hat i,\,\hat j$ are in $x,y$ directions respectively.

I can show that $\hat\theta=-\sin\theta\hat i+\cos\theta\hat j$ in geometry using the tangent chord angle theorem.

Can anyone provide an algebraic proof for the $\hat\theta$ equation?

Answer 1

The unit vector $\hat \theta$ is defined as the unit vector that is normal to the position vector $\vec r$ and points in the direction of increasing $\theta$.

Write $\hat \theta = \cos(\alpha)\hat x+\sin(\alpha)\hat y$ for some angle $\alpha$. Then, we have

$$\hat \theta \cdot \hat r=\cos(\theta-\alpha)=0\implies \alpha = \theta\pm \pi/2\tag 1$$

The ambiguity of the sign in $(1)$ is resolved when enforcing that $\hat \theta$ points in the direction of increasing $\theta$, in which case we have $\alpha = \theta +\pi/2$ and therefore

$$\hat \theta = -\sin(\theta)\hat x+\cos(\theta)\hat y$$

Alternatively, we note that

$$\begin{align} 0&=\frac{d(1)}{d\theta}\\\\ &=\frac{d(\hat r\cdot \hat r)}{d\theta}\\\\ &=2\hat r\cdot \frac{d\hat r }{d\theta}\\\\ &=2\hat r\cdot (-\sin(\theta)\hat x+\cos(\theta)\hat y) \end{align}$$

So, $\hat r$ is orthogonal to the unit vector $-\sin(\theta)\hat x+\cos(\theta)\hat y$, which points in the direction of increasing $\theta$.

Answer 2

Simple. $\hat{\theta}$ is orthogonal to $\hat{r}$ and on the xy-plane (or orthogonal to $\hat{k}$). You do that with the $\times$ vector cross product operator.

$$\hat{\theta} = \hat{k} \times \hat{r} $$