# Adding two polar vectors

Is there a way of adding two vectors in polar form without first having to convert them to cartesian or complex form?

## Answer

Here is another way forward that relies on straightforward vector algebra. Let $\vec r_1$ and $\vec r_2$ denote vectors with magnitudes $r_1$ and $r_2$, respectively, and with angles $\phi_1$ and $\phi_2$, respectively.

Let $\vec r$ be the vector with magnitude $r$ and angle $\phi$ that denotes the sum of $\vec r_1$ and $\vec r_2$. Thus,

From the definition of the inner product we have

and

Using $(1)$ and $(2)$, we find $r^2$ can be written

and thus $r$ is given by

Using $(1)$, $(3)$, and $(4)$, yields

whereupon solving for $\cos (\phi-\phi_1)$ reveals

We can easily obtain the expression for $\sin(\phi-\phi_1)$ by applying the cross product

which after straightforward arithmetic yields

Dividing $(5)$ by $(6)$ and inverting shows that

where the function $\operatorname{arctan2}(y,x)$ is described in this article.

Equations $(4)$ and $(7)$ provide the polar coordinates of $\vec r$ strictly in terms of the polar coordinates of $\vec r_1$ and $\vec r_2$. And the development of $(4)$, $(5)$, and $(6)$ did not appeal to Cartesian coordinates.

NOTE:

In a parallel development, we can express the sum of two complex numbers $z_1=r_1e^{i\phi_1}$ and $z_2=r_1e^{i\phi_2}$ in terms of their magnitudes and arguments.

First, recall that the inner product of two complex numbers is given by

where $\bar z$ denotes the complex conjugate of $z$.

Next, we let $z=re^{i\phi}=z_1+z_2$ be the sum of $z_1$ and $z_2$. The magnitude of $z$ is given by

Therefore, we have

Finally, we find the argument of $z$ by taking the inner product of $z$ and $z_1$. To that end, we write

which reveals that

whereupon inverting yields

Equations $(8)$ and $(9)$ provide the polar coordinates of $z$ strictly in terms of the polar coordinates of $z_1$ and $z_2$. Again, this development did not appeal to Cartesian coordinates.

Attribution
Source : Link , Question Author : lash , Answer Author : Mark Viola