# One spot and distance known, Second spot unknown

I know the coordinates of points E and Q, so I know their euclidean distance L. I'm looking for the point W with coordinates (a,b) related to other known values?

-
You know r as well, right? Or do you want the answer in terms of r –  The Chaz 2.0 Apr 10 '12 at 17:12
Also, that link is opening up all sorts of popups on my browser. Please use the "add image" button. –  The Chaz 2.0 Apr 10 '12 at 17:14
No, I know r . I'm looking for the a and b in terms of the others. –  user28775 Apr 10 '12 at 18:14
Assume the $E$ is the origin, for a moment. $\vec{EQ}=\binom{x}{y}=L\binom{\sin \theta}{\cos \theta}$. Then use vector addition: $\vec{EW}=\vec{EQ}+\vec{QW}$. If $E$ is not the origin, shift the result by $\vec{E}= \binom{e_1}{e_2}$. (all assuming $\theta$ is also known...) –  draks ... Apr 16 '12 at 18:06

I'll assume you know $\theta$, the coordinates $(x,y)$ of $Q$, and $r$. The angle of rotation from a vertical line through $Q$ to $\overline{QW}$ is also $\theta$, so if we take the point that is $r$ above $Q$ and rotate it by $\theta$ about $Q$, we'll get $W$. That means $$W=(a,b)=(x-r\sin\theta,y+r\cos\theta).$$