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Vector word problem: hiking

Sal solves a word problem with vectors where he finds the total straight-line distance traveled over a few days. Created by Sal Khan.

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  • marcimus red style avatar for user Isha
    What if they ask to convert the angle, which is 52 degrees, to radians? What would be the answer then? How do I change from degrees to radians?
    (2 votes)
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    • piceratops ultimate style avatar for user Just Keith
      While the math in Yamanqui's answer is correct, I'd like to suggest a slightly different approach that I have found helps students who have trouble grasping the degrees vs. radians concepts:

      First, whether in degrees or radians, find out what fraction of a circle the angle you have sweeps out. You do the by dividing that angle by a full angle (360° or 2π radians).

      Now you just have the fraction of a circle the angle sweeps out, no degrees nor radians. So, to get degrees or radians, you just multiply by 2π or 360° depending on whether you want radians or degrees.

      So, for 52°:
      52/360 = 13/90 of a circle
      We want radians, so we multiply that fraction by 2π
      13/90 × 2π = 13π/45

      For going from radians, let us take the angle ³⁄₂π
      ³⁄₂π ÷ 2π = ¾ of a circle
      ¾ × 360° = 270°
      (22 votes)
  • blobby green style avatar for user konstantinos1993
    Why is his direction at the end of day three not the direction of the vector d3 and instead he calculates it as the direction of the vector dt?
    (3 votes)
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  • winston default style avatar for user Grayson Ward
    Is "__m/s in a certain direction" considered a vector, or not?
    (2 votes)
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  • leaf blue style avatar for user Secondary Math II and III
    The answer for the total distance on the calculator was 24.28... He tells us that the questions asks for the distance rounded to the nearest tenth place, and reports the answer as 24.2. Shouldn't it be rounded to 23.3?
    (0 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      The answer on his calculator was the same as that on all of ours: 24.2074368...
      What you see as 24.28.. is a digital-style zero with a slash through it
      So, when he had to round it, he started with 24.2074368...
      He has to round to the tenth place:
      24.2074368...
      so, he checks the next digit after the tenths' place and it is a zero
      So it rounds to 24.2
      (5 votes)
  • duskpin ultimate style avatar for user Stirling Sidaway
    In the practice question pool, there are a few questions that give a vehicle's initial velocity, but then gives a different velocity when affected by wind, and asks for the magnitude and direction of the wind itself. How do I approach those problems?
    (2 votes)
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  • purple pi teal style avatar for user Nourjehan Faris
    Hi, what if I am only given the length of each distance and the degree of the angle? How would I solve for North and East in that case? For example, if the question said Keita traveled 15km [N 51degrees E] (say, in this case that the Hypotenuse equals 15km); and then gave the hypotenuse length and the angles of all the other distances; how would I get the total displacement?
    (2 votes)
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  • starky ultimate style avatar for user Ace
    Why was 19/15 the inverse tangent? Can somebody please explain the concept of inverse trigonometric functions.
    (2 votes)
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    • piceratops ultimate style avatar for user Dav Kwong
      trigonometric functions are like all other functions in having an inverse.....
      If f(x)=y, then the inverse of function f-1(y)=x, in this case....
      tan(theta)=19/15 therefore the inverse tan of 19/15 or tan-1(19/15)=theta
      The inverse of a function is basically a function that when inputted with the answer of the original function produces an answer which is equal to the input of the original function. For example, if c(x) was the inverse of f(x) and if f(2)=7 then c(7) = 2
      (2 votes)
  • blobby green style avatar for user Madhan
    I can't understand the sentence that the 'conventional tangent gives the angle between -pi over 2 and pi over 2'. Could anyone explain this sentence. Thanks in advance.
    (1 vote)
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    • piceratops ultimate style avatar for user Hecretary Bird
      In order to keep the inverse tangent a function (every input corresponds to only 1 output), we have to restrict its domain somehow. The normal tangent graph repeats itself forever and has multiple points that give the same output for different inputs. So when we take the inverse, there would be multiple outputs for one input unless we restrict the domain. A domain interval of 180 degrees captures every possible tangent value. I'm not sure if there's a mathematical reason why arctangent is restricted to -pi/2 and pi/2, besides the fact that this makes it centered at 0 and look nice.
      Because the domain is restricted like this, if we want to compute the arctangent of something with a calculator, it will only give us an answer between -pi/2 and pi/2 radians. If you know that the angle in question is in the 2nd or 3rd quadrants, you'd have to account for that yourself when finding the angle by adding 180 degrees to the value you get from the calculator.
      (2 votes)
  • aqualine seedling style avatar for user Diana Valle
    Why do we use tangent to find the angle? Why can't we take the inverse cosine or inverse sine? I've tried it on the calculator and do not get the same answer. Can someone please explain?
    (1 vote)
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  • female robot ada style avatar for user lilyprovenzano
    I did this a little differently. Instead of drawing the components of the total displacement first to the right and then up, I went up first and then to the right, so I ended up on the opposite side of the vector than Sal did. When Sal then measured the angle, he came up with 52 degrees. The 52 degree angle for me, however, since I was working on the opposite side of the vector from Sal, was at the tip of the vector, not the base. The angle I measured by the base of the vector was 38 degrees. My question is this: how do we know which angle is correct?
    (1 vote)
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Video transcript

Keita left camp three days ago on a journey into the jungle. The three days of his journey can be described by displacement, distance and direction vectors, or displacement vectors, and displacement is distance with the direction, and the vectors are d1, d2, and d3. They list them right over here. The distances are given in kilometers. How far is Keita from camp at the end of day three? So let's just think about what is happening. On day one, let's say this is his starting point. His displacement, he starts here and he goes there, but if you break it down based on, I guess if you call this direction Let me draw a little compass here. If you say that this is north, this is east, this is west, and that this is south, you can break it down by how much he went in the east direction and how much he went in the north direction. So this is saying he went seven in the east direction. One, two, three, four, five, six, seven. Seven in the east direction, and he went eight in the north direction. One, two, three, four, five, six, seven, eight. Just like that. This is seven and this is eight. Then on day two, he went six units, I guess these are kilometers, six kilometers to the east, one, two, three, four, five, six, and he went two kilometers to the north, one, two. So he ends up right over here. Six and two, and then finally day three, the component of his displacement that is to the east is two, and the component of his displacement that is to the north is nine. And so to figure out how far he is from camp at the end of day three, we just have to figure out what is his total displacement? What is the length of the vector that is the sum of all of these? So what is the length of this vector right over here? Let's call this d sub t for total, displacement d sub t for total. Displacement total, total displacement. And you can see how this is arranged, that our total displacement vector is just going to be the sum of d1, let me do those in the appropriate colors. It's just going to be the sum of d1, d2, and d3. And if you're summing these vectors, you can just add the corresponding component. So for example, the total displacement is going to be equal to the sum of the horizontal, I should say the displacement in the east direction. So it's going to be seven plus six plus two, and then in the north direction you have eight plus six, or sorry, eight plus two plus nine. And so our total displacement vector if we were to write it in this form, is going to be, this is 13 plus two, so it's going to be 15 to the east, or the component of displacement in the eastern direction is 15, and in the northern direction is 10 plus nine, 19. So that's this vector right over here. It's 15, it's component in the east is 15, and to the north is 19. Let me make that clear. So this distance right over here, or if I were to make this a triangle the length of this side of the triangle here is 19 and that's his total displacement in the northern direction, and his total displacement in the eastern direction is that 15. So what's going to be the length of our displacement vector or what's the magnitude of our displacement vector? The magnitude of our total displacement is, well it's the Pythagorean theorem. This is a right triangle. 15 squared plus 19 squared is going to be the magnitude squared. Or we could say that the magnitude is equal to the square root of 15 squared plus 19 squared, plus 19 squared. So let me get my calculator out. All right, so I have the square root of 15 squared plus 19 squared gives, 24, let's see, they want us to round to the nearest tenth, 24 point two. So 24 point two kilometers. Then they say what direction is Keita from camp at the end of day three? Round to the nearest degree. Your answer should be between zero and 180 degrees. So distance I'm assuming, this would just be the convention, they should be a little bit more precise here in their wording or a little bit less ambiguous. The convention is, the angle relative to, if we were thinking of the coordinate axes, the positive X axis, on this you could say the eastern direction, so it's really this angle right over here, let's call that theta. How could we figure out what theta is? It's part of this right triangle. We know this side has length 19 and we know this side has 15. So we know the opposite side theta and we know the adjacent side theta, so what trig function involves opposite and adjacent? Well tangent involves opposite and adjacent, so we could write that the tangent of theta is equal to 19 over 15, is equal to opposite over adjacent, 19 over 15, and to solve for theta we can just say, now we've got to make sure that our angle, if we just take the inverse tangent here, is actually the angle we're looking for, but inverse tangent, the convention is, it will give you an angle that is between negative pi over two and pi, if we're thinking in radiants, but since we're thinking in degrees, it'll give you an angle between negative 90 degrees and 90 degrees, and this angle is clearly in that. It's actually between zero and 90 degrees. So when we take inverse tangent, we know that we're going to get the right angle. Otherwise we would have to make some adjustments. So theta is going to be equal to the inverse tangent of 19 over 15, which is equal to, let's get our calculator out, let's make sure that we are in, yep we are in degree mode, and so let's take the inverse tangent of 19 divided by 15, which is equal, let's see they want us to round to the nearest degree, so 52 degrees if we round to the nearest degree. 52 degrees, which also looks right, right over here. This looks like a little bit more than a 45 degree angle, which 52 degrees seems to fit the bill, and we are done.