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October 13th, 2019, 04:05 AM   #1
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Question How can I find the sum of vectors in this triangle?

The problem is as follows:

Using the figure from below find the modulus of the resultant vector if it is known $\left\|v\right\|=3$ and $\left\|u\right\|=5$.



The alternatives given on my book are:

$\begin{array}{ll}
1.&7\\
2.&10\\
3.&14\\
4.&23\\
5.&28\\
\end{array}$

I tried all sorts of manipulations, and although it doesn't seem complicated the only thing which I could come up with was to find the third side from $\vec{v}$ and $\vec{u}$ which is given from:

$\overline{vu}=\sqrt{3^2+5^2-2(3)(5)\cos\left(120^{\circ}\right)}=7$

But that's where I'm still stuck. I'm confused at how to understand the other two vectors which are not labeled in the question. How am I supposed to sum them with the other two (I mean from the sides of the triangle). Can somebody help me with this?. An answer which would help me the most would be one trig-free and more on the geometric route if possible. This answer may require some drawing please try to include it as I'm slow at translating drawings from words.
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October 13th, 2019, 05:49 AM   #2
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Vectors.PNG
Let the two vectors without a label be $\vec{a}$ and $\vec{b}$ (as shown above).

The resultant of the four vectors can't be found without further information, so I'll assume that
$\vec{a} - \vec{v} = \vec{u} - \vec{b}$, which implies $\vec{a} + \vec{b} = \vec{u} + \vec{v}$.

Now the resultant vector is $2\left(\vec{u} + \vec{v}\right)$ and has modulus 2*7 = 14.

My assumption is equivalent to assuming that the outer segments of the dashed line in the diagram have equal length.
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October 15th, 2019, 10:12 PM   #3
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Quote:
Originally Posted by skipjack View Post
Attachment 10584
Let the two vectors without a label be $\vec{a}$ and $\vec{b}$ (as shown above).

The resultant of the four vectors can't be found without further information, so I'll assume that
$\vec{a} - \vec{v} = \vec{u} - \vec{b}$, which implies $\vec{a} + \vec{b} = \vec{u} + \vec{v}$.

Now the resultant vector is $2\left(\vec{u} + \vec{v}\right)$ and has modulus 2*7 = 14.

My assumption is equivalent to assuming that the outer segments of the dashed line in the diagram have equal length.
It seemed that the resultant could had not been found without using the cosines law isn't it?. The rest was easy, its very helpful to have a drawing to understand.
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October 16th, 2019, 08:47 AM   #4
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That's right, as the answer depends on the cosine of the given angle. You might prefer a diagram with additional labels, so that the wording "the outer segments of the dashed line" doesn't need to be used.
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