Main content

### Course: Precalculus > Unit 6

Lesson 9: Vectors word problems# 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.

## Want to join the conversation?

- 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)
- 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)

- 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)
- Because the question is about the angle from the initial camp, not where he started out on the third day. The vector from the initial camp (dt) is the sum of the three “day” vectors.(5 votes)

- Is "__m/s in a certain direction" considered a vector, or not?(2 votes)
- Yes it is. Vectors always have a magnitude (the "
*__*m/s") and a direction ("in a certain direction"). If the direction is missing then it'd be a scalar not a vector.

Hope this helped:)(4 votes)

- 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)
- 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.`2`

074368...

so, he checks the next digit after the tenths' place and it is a**zero**

So it rounds to 24.2(5 votes)

- 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)
- The resultant velocity of the vehicle is equal to the vehicle's initial velocity vector plus the velocity vector of the wind. You just need to set up a vector equation.(2 votes)

- 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)
- If you have all vectors in the magnitude (the distance) and direction (the angle) form you can convert them to their components. How to do is explained here:

https://www.khanacademy.org/math/precalculus/vectors-precalc/adding-vectors-in-magnitude-and-direction-form/v/mag-dir-vec-sums.

Once you have all vectors in component form the exercise is the same as the one described by the video.(2 votes)

- Why was 19/15 the inverse tangent? Can somebody please explain the concept of inverse trigonometric functions.(2 votes)
- 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)

- 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)
- 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)

- 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)
- You can, I tried it and they are the same answers. If you could describe the operations you're trying I'd be happy to help.(2 votes)

- 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)
- They are both correct, as long as you specify well which direction and/or angle you mean.

However, the generally accepted convention in math and physics is to measure angles from the positive x-axis, just so you know for future reference.(1 vote)

## 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.