# Expressing cartesian unit vectors in terms of plane polar unit vectors to prove that former doesn't depend on position

I understand that polar unit vectors are given by

$$e_r= cos(θ)i + sin(θ)j$$

$$e_θ=−sin(θ)i + cos(θ)j$$

How do I now express cartesian unit vectors in terms of polar unit vectors to show that they are independent of the $$r$$ and $$θ$$?

How do I now express cartesian unit vectors in terms of polar unit vectors to show that they are independent of the $$r$$ and $$θ$$?

If you express the cartesian unit vectors in terms of polar unit vectors that means that they are not independent of each other. The unit vectors $$i$$ and $$j$$ are independent and so is $$hat{r}$$ and $$hat{theta}$$. But one set is not independent of other.

To find $$i$$ in terms of $$r$$ and $$theta$$ by eliminating $$j$$ from two equation. As follows

$$cos(theta)hat{r}=cos^2(theta)hat{i}+costhetasin(theta)hat{j}$$ $$sin(theta)hat{theta}=-sin^2(theta)hat{i}+cos(theta)sin(theta)hat{j}$$ substracting:

$$hat{i}=cos(theta)hat{r}-sin(theta)hat{theta}$$ similear procedure for $$hat{j}$$.

If you want to show the linear independence of $$hat{r}$$ and $$hat{theta}$$ then you can show they are perpendicular to each other: $$hat{r}cdothat{theta}=0$$ and for unit vectors $$hat{r}cdothat{r}=1=hat{theta}cdot hat{theta}$$.

Correct answer by Young Kindaichi on January 6, 2021