How can I find the resultant and norm from pair of vectors passing in half a circle?

Jun 2017
Lima, Peru
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:

1.&2\,\mu m\\
2.&6\,\mu m\\
3.&2\sqrt 3\,\mu m\\
4.&4\, \mu m\\
5.&\sqrt 3\, \mu m\\

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.
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Forum Staff
Dec 2006
In the above diagram, I've constructed some equilateral triangles whose sides are of the same length as the radius of the circle. Note that the lines GC and AG correspond to your vectors. Hence, you are effectively asked for the length of AC, which you can see is twice the length of an altitude of one of the equilateral triangles, so it's easily calculated to be 2√3 μm. You will first need to find a few angles, so that you can show that my diagram is valid. You might choose to add other lines to the diagram or omit the line CD.