Adding equal and opposite vectors in polar coordinates without transforming into cartesian Suppose that you have two force vectors, $\vec{F_{1}}$ and $\vec{F_{2}}$ in polar coordinates such that they have equal magnitude and opposite angles, $\theta$ as defined below.
$$ \vec{F_{1}}  = r\hat{r}+\frac{\pi}{4}\hat{\theta}$$
$$ \vec{F_{2}}  = r\hat{r}+\frac{5\pi}{4}\hat{\theta}$$
We can assume from symmetry that the total force, $\vec{F_{Total}} = 0$.
Is there a way to show this using only polar coordinates and without resorting to cartesian coordinates?
 A: Yes, it is possible to add vectors without a Cartesian
representation, i.e. represented by two-dimensional r/theta coordinates.
One simply can draw an 'r' length line segment in the 'theta'
direction, and, at its apex, append a second 's' length
vector in the 'theta2' direction.   Drafting machines like this 
with protractor and scale components are an easy way to
accomplish this, and make it easy to decode the resultant
sum.
As for doing it numerically, I'd stick with Cartesian coordinate transformation.   It's  not hard.
A: The force components are:
$\def \b {\mathbf}$
\begin{align*}
  &\b F_1=r\,\b e_r+\frac{\pi}{4}\b e_\theta\\
 &\b e_r=\begin{bmatrix}
           \cos(\theta) \\
           \sin(\theta) \\
         \end{bmatrix}\quad,
    \b e_\theta=\begin{bmatrix}
           -\sin(\theta) \\
           \cos(\theta) \\
         \end{bmatrix}    \quad ,\b e_r\perp\b e_\theta 
  \end{align*}
with $~\theta\mapsto -\theta$
\begin{align*}
  &\b F_2=r\,\b e_r+\frac{5\pi}{4}\b e_\theta\\
 &\b e_r\mapsto\begin{bmatrix}
           \cos(\theta) \\
           -\sin(\theta) \\
         \end{bmatrix}\quad,
    \b e_\theta\mapsto\begin{bmatrix}
           \sin(\theta) \\
           \cos(\theta) \\
         \end{bmatrix}    \quad ,\b e_r\perp\b e_\theta
  \end{align*}
thus $~\b F_1-\b F_2\ne \b 0$

