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## Precalculus

### Course: Precalculus>Unit 6

Lesson 9: Vectors word problems

# Vector word problem: resultant force

When two different forces act on the same object, we can find the resultant force acting on the object by adding the two separate forces. In this example, we find a resultant force vector using geometry, specifically the laws of sines and the laws of cosines. Created by Sal Khan.

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• If that's just introduction of how we can use other trigonometric approaches to solve the same problem, that's good, but otherwise i can't see why it can't be solved by the same exact formulas we used before, assuming the first vector just has the zero angle. Isn't it much simpler anyway? • I applied a completely different method and got the correct answer when rounded. I got the exact same answer as Sal for the magnitude, but my angle is slightly different. Shown below is my work.

I turned this into a coordinate plane and assigned the force with 3 Newtons a value of (0,3). I then rotated 100 degrees to the right and gave the force with 5 Newtons a magnitude of 5 and an angle -10 degrees.

The two forces can be written in magnitude and direction form as :
t=(3cos(90), 3sin(90))
f=(5cos(-10), 5sin(-10))

I then added those two together to get:

t+f= (3cos(90)+ 5cos(-10), 3(sin90)+ 5sin(-10))

In order to get the most accurate results as possible, I squared I plugged in the values this way into the Pythagorean theorem as well.

||t+f||=sqrt((3cos(90)+ 5cos(-10))^2 + (3sin(90)+ 5sin(-10))^2)

||t+f||= 5.36 or 5.4 when rounded. So far so good.

In order to find the direction relative to the first magnet, I had to find theta and add 10 degrees to it.

So I did

theta= arctan((3sin(90)+5sin(-10)) / ((3cos(90)+5cos(-10)))
theta= 23.4

I then added 10 degrees to get 33.4 degrees which when rounded, gives us 33 degrees.

But Sal got 33.2 degrees as his theta. Why am I getting 33.4 degrees as my theta? • I just treated the vector with a magnitude of 5 Newtons as the x-axis. This made the math easier and I still got the same answer. Is this a better way to do the problem that Sal didn't think about?   