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October 13th, 2019, 04:08 AM  #1 
Senior Member Joined: Jun 2017 From: Lima, Peru Posts: 188 Thanks: 5 Math Focus: Calculus  Is there a way to obtain the modulus of desired sum from these vectors?
The problem is as follows: Find the modulus of the resultant vector which are seen in the quadrilateral shown below. Consider M and N to be mid points and \overline{MN}=20 The alternatives given on my book are: $\begin{array}{ll} 1.&20\\ 2.&40\\ 3.&60\\ 4.&80\\ 5.&50\\ \end{array}$ I'm stuck at this problem. First off, I'm not sure whether the schematic is incomplete (I've copied it down from my book). The reason for what I believe that seems incomplete is that I don't why $\overline{MN}=20$ is not in the picture. Will it affect the answer? I don't know what was intended by the one who posed this question. I really need guidance with this question as I don't know how on earth I could manipulate the existing vectors to obtain a logical sum. Can somebody help me with this using geometry or something trigfree?. As I mentioned, this problem seems kind of incomplete and I believe a drawing or some sketch which could accompany the answer would be much better to aid my understanding. Last edited by skipjack; October 13th, 2019 at 07:06 AM. 
October 13th, 2019, 07:02 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 21,124 Thanks: 2332 
Let the lower dashed line be ANB. As $\vec{u} + \vec{v} = \vec{MA} + \vec{MB}$ (I hope you can see why), the modulus of $\vec{u} + \vec{v}$ is twice the length of MN. 
October 15th, 2019, 10:15 PM  #3 
Senior Member Joined: Jun 2017 From: Lima, Peru Posts: 188 Thanks: 5 Math Focus: Calculus  A drawing please? As I mentioned above, my first impression was that this problem was incomplete. But it doesn't seem that. Can you add some drawing to aid me understand the justification for this answer? I don't know exactly how to get to that equality that you had shown; if you could draw the arrows over my drawing or make one, I could understand. Can you help me with that? Last edited by skipjack; October 16th, 2019 at 09:30 AM. 
October 16th, 2019, 09:35 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 21,124 Thanks: 2332  VectorSum.PNG As M is the midpoint of CD, $\vec{MC} = \vec{MD}$. $\vec{MA} + \vec{MB} = \vec{MD} + \vec{u} + \vec{MC} + \vec{v} = \vec{u} + \vec{v}$ As MN is the relevant median of triangle MAB, $\vec{MA} + \vec{MB} = 2\vec{MN}$. 

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desired, modulus, obtain, sum, vectors 
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