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October 13th, 2019, 04:10 AM  #1 
Senior Member Joined: Jun 2017 From: Lima, Peru Posts: 188 Thanks: 5 Math Focus: Calculus  How to obtain the resultant and its modulus from a group of vectors in a circle?
The problem is as follows: Find the modulus of the resultant from the vectors shown in the picture from below: The alternatives given on my book are: $\begin{array}{ll} 1.&10\,\textrm{inch}\\ 2.&20\,\textrm{inch}\\ 3.&10\sqrt{3}\,\textrm{inch}\\ 4.&20\sqrt{3}\,\textrm{inch}\\ 5.&60\,\textrm{inch}\\ \end{array}$ For this problem the only thing I could come up with is described in my attempt seen in the figure from below: I thought that it was easy to form a closed polygon and from there I could obtain a sum like this, hence the resultant: $\vec{r} = \vec{u} + \vec{v} + \vec{w} + \vec{x} + \vec{y} + \vec{z}$ $\vec{x}+\vec{y}+\vec{z}=\vec{w}$ $\vec{w}+\vec{v}=\vec{u}$ $\vec{r}=\vec{u}+\vec{u}+\vec{w}$ For the sake of brevity, I'm omitting units. Since it is known the radius is $\textrm{10 inch}$ then: $\vec{w}=20$ and from there it can be inferred that: $\vec{u}=10$ Then: $\vec{r}=10+10+20=40$ But my answer doesn't appear in the alternatives. Am I doing something wrong? Could it be that Am I misinterpreting the concepts?. I'd like somebody could take a look into this as I'm confused with vectors. Since I believe an auxiliary drawing can be required to aid understanding of the answer I hope that somebody could help me if there is some sort of geometrical manipulation which can be done to solve this problem. I'd like to note that apparently $\textrm{O}$ is the center of the circle. Last edited by skipjack; October 13th, 2019 at 06:00 AM. Reason: added information. 
October 13th, 2019, 06:49 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 21,124 Thanks: 2332  
October 15th, 2019, 10:07 PM  #3 
Senior Member Joined: Jun 2017 From: Lima, Peru Posts: 188 Thanks: 5 Math Focus: Calculus  Uh I'm still confused Sorry skipjack. But I had to take an exam (on this subject) so I couldn't reply early. Anyways, can you please tell me exactly what step I did wrong?. I don't know the graphical justification for your answer. Perhaps if you could show me this part I can "see" where's the resultant. Can you help me with that because I'm slow at picturing these things on my head solely reading words. 
October 16th, 2019, 08:15 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 21,124 Thanks: 2332 
You first went wrong with $\vec{w}=20$, as it's the vector's modulus that is 20. Your $\vec{r}=\vec{u}+\vec{u}+\vec{w}$ was correct, but $\vec{r}=10+10+20=40$ was wrong. Hence $\vec{r} = 2\vec{u} + \vec{w}$ would be correct. As $2\vec{u}$ and $\vec{w}$ are vectors of equal modulus, but there's an angle of 60 degrees between them, adding them by use of the parallelogram method produces a resultant that is twice a median of a certain triangle, which is equilateral in this case. See the diagram below for the addition of any two vectors by the parallelogram method. As the diagonals of a parallelogram bisect each other, PR = 2PT and PT is a median of triangle PQS. For the current problem, the median's length can be found by using Pythagoras. In the general case, trigonometry can be used. ParallelogramMethod.PNG 
October 21st, 2019, 09:18 PM  #5  
Senior Member Joined: Jun 2017 From: Lima, Peru Posts: 188 Thanks: 5 Math Focus: Calculus  Quote:
 

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circle, group, modulus, obtain, resultant, vectors 
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