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Suppose I have a vector $\vec{v}_1$ going from the origin to a point a distance $r$ from the origin to a circle that is centered on the origin. Then I can write this vector as

$$\vec{v}_1 = r\hat{r} + \theta_1\hat{\theta}$$

Suppose there is another vector, again going a distance $r$ from the origin to a point on the circle, which I can write as

$$\vec{v}_2 = r\hat{r} + \theta_2\hat{\theta}$$

Now if I subtract these two vectors, I get a vector going from the head of $\vec{v}_1$ to the head of $\vec{v}_2$:

$$\vec{v}_2 - \vec{v}_1 = r\hat{r} - r\hat{r} + \theta_2\hat{\theta} - \theta_1\hat{\theta} = (\theta_2 - \theta_1)\hat{\theta}$$

Then $\hat{\theta}$ is co-linear with $\vec{v}_2 - \vec{v}_1$.

Suppose there is another vector $\vec{v}_3$ on the circle:

$$\vec{v}_3 = r\hat{r} + \theta_3\hat{\theta}$$

Then I can repeat the above argument to say that $\hat{\theta}$ is co-linear with $\vec{v}_3 - \vec{v}_2$. But then $\hat\theta$ is co-linear with both $\vec{v}_2 - \vec{v}_1$ and $\vec{v}_3 - \vec{v}_2$ and so $\hat{\theta} = \vec{0}$.

What is wrong with my reasoning?

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