geometryvectors
I have 2 points, A + B, with vectors from the origin a and b. The vector from A to B is
c = b – a
a and b are defined in polar coordinates with
a $ = (r_a,\theta_a) $, b $ = (r_b,\theta_b ) $
I want to know how to define the angle c makes with the horizontal in terms of $\theta_a $ and/or $\theta_b$.
Hint:
Grow the circle $A$ to a radius of $r_A+r_B$ while you shrink that of $B$ to $0$. The common tangent keeps the same direction. Then you have a right triangle with sides $P_AP_B$ and $r_A+r_B$. Now you add the direction angle of $P_AP_B$ and the angle of the triangle.
So we have $AB=(-3,5,-1)$ and $OA=(2,2,3)$. Now use, for example, the formula
$$\frac{a\cdot b}{|a||b|}=\cos\theta,$$ where $\theta$ is the angle between $a$ and $b$.
Edit: I think the confusion here is what we really mean by angle between two vectors. Since vectors "float" in space, we can imagine placing them so that their tails touch. Then the angle between these two physical interpretations of vectors is defined as the angle between $a$ and $b$ as vectors. There is no ambiguity about where the vector is pointing because we place them tail-to-tail so both arrows always point away.
Here is a pretty bad picture I made in photoshop. Even though b "points away" from the angle, it doesn't really affect anything.
Best Answer
If it is an isosceles triangle, $\angle A=\angle B=90-\frac{\theta_a}{2}$
Extending $c$ down to the horizontal makes a triangle with angles $\theta_b,180-(90-\frac{\theta_a}{2}),\theta_c$. With $\theta_c$ being the angle you are after. It can now be expressed in terms of $\theta_a$ and $\theta_b$