# Math Help - [SOLVED] Easy Vectors Question

1. ## [SOLVED] Easy Vectors Question

Find the magnitude and direction of the equilibrant (resulting force) of each of the following forces: 16 N and 10 N acting at an angle of 10 degrees to each other.
Please make it as visual as possible... Thanks! (best answerer gets a thanks ^^)

2. Originally Posted by x y z
Find the magnitude and direction of the equilibrant (resulting force) of each of the following forces: 16 N and 10 N acting at an angle of 10 degrees to each other.
There are different ways to solve this question:

1. Draw the parallelogram of forces. Use the Cosine rule to calculate the length of the larger diagonal. Keep in mind that the adjacent sides of the parallelogram include an angle of 180° - 10° = 170° :

$F_r = \sqrt{10^2+16^2-2 \cdot 10 \cdot 16 \cdot \cos(170^\circ)}$

2. Use vectors:

Let $\overrightarrow{F_1} = (10,0)$ and

$\overrightarrow{F_2} = 16(\cos(10^\circ), \sin(10^\circ))$
Then

$\overrightarrow{F_r} = (10,0) + 16(\cos(10^\circ), \sin(10^\circ)) = (10+16 \cos(10^\circ), 16 \sin(10^\circ))$

Now calculate the magnitude of $\overrightarrow{F_r}$ :

$| \overrightarrow{F_r} | = \sqrt{\left(10+16 \cos(10^\circ) \right)^2 + \left(16 \sin(10^\circ) \right)^2}$

3. Thanks.
But how do I then get the direction of the vector?

4. Originally Posted by x y z
Thanks.
But how do I then get the direction of the vector?
If you have the vectors $\vec u$ and $\vec v$ then the included angle is calculated by:

$\cos(\angle(\vec u , \vec v)) = \dfrac{\vec u \cdot \vec v}{|\vec u| \cdot |\vec v|}$

$
\cos(\angle(\overrightarrow{F_r} , \overrightarrow{F_1}) = \dfrac{(10+16 \cos(10^\circ), 16 \sin(10^\circ))
\cdot (10, 0)}{2\cdot \sqrt{80 \cdot \cos(10^\circ)} \cdot 10}$

After your calculator has cooled down a little bit you'll get:

$\angle(\overrightarrow{F_r} , \overrightarrow{F_1}) \approx 6.157^\circ$

Keep in mind that you now have calculated the angle which is included by $\overrightarrow{F_r} \text{ and } \overrightarrow{F_1}$

5. That's what I got too, but the answer says that it's 174 degrees to the 10N vector. Why did they subtract from 180 degrees?

6. Originally Posted by x y z
...
Please make it as visual as possible... ...
Originally Posted by x y z
That's what I got too, but the answer says that it's 174 degrees to the 10N vector. Why did they subtract from 180 degrees?
I've attached the sketch of the situation ... and honestly I can't explain where the result in your textbook comes from, sorry!

7. Wow, thank you so much! <3

8. Originally Posted by x y z
That's what I got too, but the answer says that it's 174 degrees to the 10N vector. Why did they subtract from 180 degrees?
I finally found out where the answer in your textbook comes from:

The most important word in your question is equilibrant (resulting force).

If you combine the given forces to a resultant force and you should find a third force such that the complete system is in equilibrium then the result must be a force with the magnitude of $\overrightarrow{F_1}+\overrightarrow{F_2}$ and the opposite direction. That is $-\left( \overrightarrow{F_1}+\overrightarrow{F_2} \right)$.

I've attached a sketch. You easily find the angle in question.