Adding equal and opposite vectors in polar coordinates without transforming into cartesian

Question

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?

To state this another way for future readers: If I have an object at the origin of the coordinate system and two forces are applied one at 45 degrees and one at 225 degrees the net force on the object will be zero if the magnitudes of the forces are the same

Answer 1

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.

Answer 2

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 $~\b F_1-\b F_2\ne \b 0$