I've been walking in circles (no pun intended) with this problem. It states as follows:
A certain sugar is analyzed at an optical laboratory. The tecnician passes two beams in the visible spectra one orange and the other lightblue. These describe the vectors labeled A and B (see the figure from below). Find the modulus of the resultant vector if it is known the radius is 2 micrometers.
The alternatives given on my book are:
$\begin{array}{ll} 1.&2\,\mu m\\ 2.&6\,\mu m\\ 3.&2\sqrt 3\,\mu m\\ 4.&4\, \mu m\\ 5.&\sqrt 3\, \mu m\\ \end{array}$
Typically I would try to show some attempt to solve this problem but here I'm stuck at the very beginning. I tried all sorts of manipulations with the vectors. But I couldn't really reach to a logical arrangement to obtain a resultant. In other words I got tangled with too many arrows. Can somebody help me with this?.
If possible I'd appreciate a graphical and simple approach with less algebraic manipulations if possible. Please try to accompany the explanation with some sort of drawing so I can see what's happening because I feel this kind of question does really need that, otherwise I'll not be able to "understand" what's going on.
You are after $\left\lVert\vec A+\vec B\right\rVert$. You can get it using the fact that$$\left\lVert\vec A+\vec B\right\rVert^2=\left\lVert\vec A\right\rVert^2+\left\lVert\vec B\right\rVert^2-2\cos(\theta)\left\lVert\vec A\right\rVert\left\lVert\vec B\right\rVert,$$where $\theta$ is the angle between $\vec A$ and $\vec B$. See the picture below: