Writing vectors in polar form

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?