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Vector components from magnitude & direction: word problem

Given the magnitudes and directions of the forces two people are applying to a box, find the overall force that is applied in the direction of the target. Created by Sal Khan.

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  • blobby green style avatar for user Harikesh
    I am not convinced with sal's explanation and this is how i feel. Please check what is wrong with this reasoning.

    If we plot same scenario on 2D graph assuming box is at origin 0, 0 and target is at some positive number on y axis then :
    Vertical component of vector a becomes cos(35) * 330 = 270 (approximated) and horizontal coponent become sin(35) * 330 = 189.
    Since horizontal component is acting away from origin in negative direction, we can represent vector a = -189i + 270j.

    Similarly vertical component of vector b becomes cos(15) * 300 = 290 (approximated) and horizontal coponent become sin(15) * 300 = 78.
    Since horizontal component is acting away from origin in positive direction, we can represent vector b= 78i + 290j.

    Note that vertical components of both vectors are acting towards target which we have assumed is in positive direction, so they are positive.

    Hence resultant vector of and b can be given as a + b = (-189+78)i + (270+290)j = -111i + 560j.

    So resultant vector is not going in the direction of target, rather it is moving away from target in negative direction horizontally.

    To calaculate magnitude of resultant vector call it c.
    c = sqrt(-111*-111 + 560*560) = 570.89 Newton

    What is wrong with it ?
    (15 votes)
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    • male robot johnny style avatar for user Thomas B
      The problem is that we are only concerned with the the vectors in the direction of the target (perhaps the box is on rails), so all we are concerned with are the 270 and 290 force vectors and we can consider the remaining force (the i component in your case) as wasted energy.
      (32 votes)
  • duskpin ultimate style avatar for user Jett Thistle
    What is the whole "sub x" thing for?
    (0 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      a "sub x" means the amount of force in the x direction from vector a

      b "sub x" means the amount of force in the x direction from vector b

      Although this video didn't dwell on it, a "sub y" would be the amount (magnitude) of force in the y direction from vector a and so on
      (14 votes)
  • female robot amelia style avatar for user Luca Valenti
    I know that the result is the same, but if we have considered the dashed line as our horizontal axis, wouldn't vector b have an angle of -15?
    (5 votes)
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    • leafers seedling style avatar for user CCDM
      Yes, but cos(-15) = cos(15).

      Whats more interesting to me is that there is a force that is perpendicular to the line between the box and the target which could result in the box never reaching the target. Person A exerts a perpendicular force which is 330sin(35)= 189N and person B exerts a perpendicular force 300sin(-15)= -78N. Notice these two are in opposite directions. Since the perpendicular force of A is higher than the perpendicular force of B, we know the box will never reach the target. It will veer off to the left of the target.

      So how much does person A need to back off on pushing so they exactly cancel out the components perpendicular to the line of motion? If we change person A force from 330N to F and we want all the forces perpendicular to the desired line of motion to be zero, we get Fsin(35) + 300sin(-15) = 0 or F = 135.4N. So the moral of the story is, if Person A wants to push at a angle more off course from target then person B, dont push so hard. Let person B do most of the work :)
      (5 votes)
  • blobby green style avatar for user yan chen
    Would the box really move towards the target exactly? Or it might end up at some other spot?
    (3 votes)
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    • leaf blue style avatar for user jwinder47
      It will definitely end up at some other spot! The problem Sal is interested in is not where the box will end up, or in which direction it moves, but rather: what forces are exerted on the box in the precise direction of the box . This comes down to calculating the component of each force in this specific direction. The final question is: who exerts more force directly toward the box ? (This question is asked at around .)
      (5 votes)
  • male robot johnny style avatar for user Phil
    How do I find the magnitudes of two vectors, a, b, if I'm only given just the angles for a and b, and also given the magnitude of a third vector, c, s.t. a+b+c=0 ?

    I've been running in circles trying to figure this out..
    (5 votes)
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  • aqualine ultimate style avatar for user Apollo
    Will the box move in the given direction ( if it is not moving in that direction currently ) ?
    (4 votes)
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  • purple pi teal style avatar for user Tyler Jang
    Why does Sal use a "sub x" instead of ||a "sub x"|| at ? Is there a specific benefit to not using vector notation or is it simply a matter of preference?
    (2 votes)
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  • duskpin sapling style avatar for user Arnab Dey Khan
    According to the parallelogram law of addition of two vectors:
    vector R = vector A + vector B
    => ||vector R|| = sqrt(A^2 + B^2 +2AB cos theta ) [ where theta = the angle bet. vec. A and vec. B]
    But, as per this formula the answer comes 571.115 N, why this difference ?
    (2 votes)
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  • blobby green style avatar for user Aryeh Sehayik
    Essentially what you did, in other words to find the combined force (270 +290) is add the vector a plus the vector b and then find the magnitude of the new vector?
    (0 votes)
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    • piceratops ultimate style avatar for user Christopher
      I don't think that's quite right: adding the two vectors themselves wouldn't (necessarily) produce a vector pointing to the target.

      What has been done in the video is to find the vector components that lie on a line going through the target and then adding those components. See what Sal says at around .
      (6 votes)
  • blobby green style avatar for user Mosie.neakngen12345
    Can someone please help me with this question?
    Two winches are being used to pull out a bogged car. One winch is on the driver's side at an angle of 32° to the forward direction and the larger winch is on the passenger's side at an angle of 39° to the forward direction. The first is exerting a force of 6000N and the other exerting a force of 9000N. What is the total force on the car, and in what direction does it act?
    (1 vote)
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    • leaf green style avatar for user cossine
      Just resolve the forces in x and y direction then use pythagora find the total force which should give you 12335.37 N at a direction of 11.62 degrees(using trigonometry) to the passenger side given i have not made a mistake
      (3 votes)

Video transcript

Voiceover: Let's say that you have two folks that are trying to collectively push a box across the snow towards a target, so this is where the box is, right over here and this is the target, right over here. Let me write that, that is the target. That's where they're trying to get the box, and person A, because for some reason, they can't push it from exactly behind the box, maybe there's not a good footing there, or I guess if they pressed there, maybe the box squeezes a little bit, so person A has to push in a direction that's not exactly going in the direction of the target, so they push in a direction that looks like this, and so let me show you this vector I'm drawing, essentially represents the force that they're exerting. This is their force vector. This is person A's force vector. So this is person A's force vector, and we know the length of this vector, or another way to think about it, the magnitude of vector A is 330 newtons, 330 newtons. And let's say person B, once again, because they can't push exactly in the direction of the target, maybe the box is really soft right over here, person B is pushing at this angle right over here, so that right over there, that vector is the force that person B is pushing onto it and the direction, and the magnitude of that force, of that vector of person B's pushing, is 300 newtons. And we know their angles that those make with the direction of the target. So if this is the direction of the target right over here we know that this is a 35 degree angle, and we know that this is a 15 degree angle. Now what I want you to do, is pause this video and think about how much of each of their force is going in the direction of getting the box towards the target, and then who is actually exerting more force in that direction. We see that person A is exerting a total, a higher magnitude of force in this direction than person B is doing in that direction. 330 newtons versus 300, but who's helping the box go more in that direction? And by how much more? And also, what's the total force now pushing the box in that direction? So I'm assuming you had a go at it, and the key here is to find the component of each of these vectors, the magnitude of each of these vectors in this direction, in the direction towards the target. And so, let's look first at vector A. So vector A looks like this, and I'll just draw them separately just so we can clean it up a little bit. Vector A looks like this. We know its magnitude - the magnitude of vector A is equal to 330 newtons, and if we say that the target is in this direction right over here, so that's the target is some place out here, we've already been told that this is 35 degrees. So what we really want to do is find the component, the magnitude of the component going in that direction right over there. The way we can do that is just with our traditional trig functions. This right over here is a right triangle. We are looking for this side right over here. We have - let me just call that A sub X. And we already know that the magnitude of A is 330 newtons, so the magnitude - let me just say that the magnitude - let me just write it this way so we don't confuse. So let's just say the magnitude of the vector in the X direction, so this vector right over here, we can write like this. The magnitude of that, we'll just write it without the vector notation. So how can we think about that? Well we know cosine is adjacent over hypotenuse, so we could write cosine of 35 degrees is equal to the length of the adjacent side. That would just be A sub X, without the vector over it. We're just saying that that's the magnitude of vector A sub X, over the magnitude of vector A, over 330 newtons, or we could say that A sub X is equal to 330 times the cosine of 35 degrees. And we can make the exact same argument for B. Vector B, so let me draw it like this. Vector B, you could, maybe I'll draw it like this just to make it a little bit - let me do it a little bit different. If this is the direction to the target, so once again, I'll just draw a horizontal line for that. Then relative to that, vector B looks something like this. Vector B looks something like that, so that is vector B. B sub X, in the direction of the target - so we would drop a perpendicular like that. This would be - this right over here would be the vector B sub X, and so what is the magnitude of B sub X going to be equal to? Now we could say that the magnitude of B sub X - we'll just call that B sub X without the vector notation, same exact logic. This is 15 degrees. Cosine of 15 degrees is going to be the length of the adjacent side over the length of the hypotenuse. So we already know that the length of the hypotenuse is 300 newtons. So we could write that cosine of 15 degrees is equal to B sub X, length of the adjacent side over the length of the hypotenuse. Or that B sub X is equal to 300 times cosine of 15 degrees. So let's get our calculator out and let's calculate what these things are. So, let's see, we have 330 times cosine of 35 degrees, gets us to 270 newtons. So that's A sub X is 270 newtons, and B sub X is 300 times cosine of 15 degrees. We get 289.777, so what we see is, even though B's magnitude is less than A's magnitude, the component of vector B going the direction of the target is actually larger than the component of vector A going in it. So if we were rounding to the nearest newton, this right over here, the magnitude of this vector right over here, B sub X, that is, if we round to the nearest newton, 290 newtons. So this is approximately 290 newtons length, or I guess you could say magnitude, while this one is a little bit shorter - it's a little bit shorter. We saw if we round to the nearest newton, it's about 270 newtons. The length of this one is 270 newtons, approximately. So if you were to say, how much more is person B pushing in the direction that we care about, it's about - well, if we want to be a little more precise, we can subtract the two, so we can take 300 cosine of 15, minus 330 cosine of 35, and we get about 19.5 newtons difference. The blue, person B, is contributing 19.458 newtons more in that direction towards the target than person A is. But if we wanted to talk about, what is the total force going in that direction, then we would take the sum of these two things. So we would - the total force in that direction is going to be 560 newtons if we round to the nearest newton. So if you add this blue component to this magenta component, you get this one right over here, which is 560 newtons. So this whole vector right over here, it's magnitude - so I could write that as the magnitude of A sub X, plus B sub X, which is the same thing as A sub X, plus B sub X - I already said these are the equivalent of the magnitude of each of these vectors is equal to - I could write approximately equal 560 newtons.