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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.

## Want to join the conversation?

- 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?(8 votes)
- You can, but the angle of the resulting vector gives out an angle that differs positively by ten degrees; perhaps it's due to the reference frame.(0 votes)

- 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**?(2 votes)- Your work is correct.

Sal got a rounding error when calculating 𝜃,

because he plugged ||𝑭|| ≈ 5.4(3 votes)

- 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?(3 votes)
- Doesn't law of sine also work?(1 vote)
- No ,it doesn't apply on resultant force, it is applied on resolution of a force(2 votes)

- I got the exact same answer, but I set the vector with a force of 5N as a 0 degree angle, and the force of 3N as a 100 degree angle. If the angle between the two vectors is given, shouldn't this strategy always work?(1 vote)

## Video transcript

- [Instructor] We're
told that a metal ball lies on a flat, horizontal surface. It is attracted by two
magnets placed around it. We're told that the first
magnets force on the ball is 5 N. We're then told the second
magnets force on the ball is 3 N in a direction that is 100 degree rotation
from the first magnets force. And we can see that drawn here. This is the first magnets force, it's 5 N. And then the second magnets force is 3 N at 100 degree angle, 100 degree rotation from the first magnets force. Now they're asking us a
few interesting questions. What is the combined strength
of the magnets' pulls? And then they also say, what is the direction of
the magnets' combined pulls, relative to the direction
of the first magnets pull? So I encourage you, pause this video and have a go at this on your own before we work through this together. All right, now let's work
through this together. So they're really saying, if I take the sum of these two vectors, what is gonna be the
resultant force vector? What is going to be the magnitude of that result in force vector? And what is its direction going to be? There's two ways we could approach this. We could break it down
each of these vectors into their respective components, and then add the respect of components, and then from that figure
out what the magnitude and direction is. And we do that in other videos, or we could take the geometric approach. So that's what we're gonna do here. And to help us with that, we're gonna use what we've
called the Parallelogram rule which is really the same idea as the head to tail addition of vectors. I can take the 3 N vector, I can shift it over, so its tails at the
head of the 5 N vector. It would look something like this. So this is 3 N right over there. And then I could also
go the other way around. I could take the 3 N vector first, and take the tail of the 5 N vector at the head of the 3 N vector
and shift it like this. You can add an either direction and either way you look at it, when you start at the tails
and you get to the head of the second vector, you're going to have a resultant force that looks like this, which is the diagonal
of this parallelogram. So there we go. And let me just call that our
force vector right over there. So if we can figure out
the length of this line of this diagonal right over here, that would be the magnitude
of this force vector. Now, how can we do that? Well, let's just think geometrically what else we can figure out
about what's going on over here? This is a parallelogram. So if this is 100 degree
angle right over here, this angle right over here is
also going to be 100 degree. And we also know that
these two opposite angles are also gonna have the same
measure right over here. And we also know that the
sum of all of the angles in a quadrilateral are
going to be 360 degrees. So these two make up 200 degrees, we have 160 degrees left
that have to be split between that one and that one. So we know that this is 80 degrees, and we know that this is 80 degrees. Well, how does that help us? So we know the length of this brown side, we know the length of
this side right over here. We know the angle between them, and what we're trying to do, is figure out the length of
the side opposite this angle, opposite this 80 degree angle. And some of you might remember
the Law of Cosines here. And the Law of Cosines I always imagine it as an adaptation of the
Pythagorean theorem, so that we can deal with
non-right triangles. And the Law of Cosines will tell us that the magnitude, I'll
just write it over here, the magnitude of this vector, which is the length of this diagonal, is going to be equal
to the square root of, we're going to have this side squared, so let me write 3 squared, plus this side squared, plus 5 squared minus 2 times this side. So times 3 times that side. So times 5 times the cosine of 80 degrees. And so let's get our calculator
out to calculate that. I'll start with taking
the cosine of 80 degrees, then I'm just gonna multiply
that times looks like 30. So times 30 is equal to that. Let's put it a little negative there. And then to that, I'm going
to add 25 and 9, which is 34. So plus 34 is equal to that. And now I just take the
square root of all of that. And they tell us to round our
answer to the nearest tenths. So I can round this to approximately 5.4. So this is approximately 5.4 N. Now they say, what is the direction of the magnets combined poles relative to the direction
of the first magnets pole? So really what we want to do, is figure out this angle right over here, let's call that data. Well, we know what the length
of the side opposite is. So maybe we could use the Law of Sines. The Law of Sines would tell us that the sine of theta over the length of the side opposite to it is going to be equal to, let's
pick another angle we know, sine of this angle. Sine of 80 degrees over the length of the
side opposite to it. And so this is approximately 5.4. And so if we wanna solve a for theta, we can multiply both sides by 3. So we're going to get sine of theta, I'll just stay in this
purple color for simplicity, is equal to 3 times sine of 80 degrees divided by 5.4. And then we could say that theta is equal to the inverse sine
of all of this business. Three sine of 80 degrees over 5.4. So we're going to take 80
degrees, take the sine of it, we're going to multiply that by 3, divide that by 5.4, that equals that, and then I'm going to take the
inverse sine of all of that. And they want us to round
to the nearest integer. So that's approximately 33 degrees. When you do the Law of Sines, it's possible that you're also
dealing with an obtuse angle. And when you do all of
this, you get the acute one, and then you would have
to make an adjustment. But that's not what we're dealing here, so we know that this data is approximately equal to 33 degrees. So we know the magnitude of the force and we know that it forms an angle of approximately 33 degrees with the direction of the
force of that first magnet.