Video: Solving Word Problems by Adding Quantities Represented as Vectors in Magnitude and Direction Form

A woman started walking from home and walked 4 miles east, 7 miles southeast, 6 miles south, 5 miles southwest, and 3 miles east. How far did she walk in total? If she walked in a straight line back home, how far would she have to walk? Give your answer correct to three decimal places if necessary.

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Video Transcript

A woman started walking from home and walked four miles east, seven miles southeast, six miles south, five miles southwest, and three miles east. How far did she walk in total? If she walked in a straight line back home, how far would she have to walk? Give your answer correct to three decimal places if necessary.

In this question, we’re given some information about a woman who’s walking from her home. First, we’re given her journey split up into five parts. And in each of these parts, we’re told the distance she traveled and the direction she traveled in terms of compass directions.

The first thing we’re asked to do is use this information to determine how far that she walked in total. And in fact we can do this straight away. If her journey splits up into five steps and we’re told the distance traveled in each of these five steps, then the total distance she walked will just be the sum of these five values. Therefore, the total distance that she walked will be four miles plus seven miles plus six miles plus five miles plus three miles. And we can calculate this. It’s equal to 25 miles.

But this isn’t the only part of this question. The second part of this question wants us to determine how far she would need to walk if she walked back to her home in a straight line. And, if necessary, we’re told to give our answer to three decimal places.

In fact, there’s a few different ways we could try and answer this question. However, it’s usually a good idea to sketch the path that she traveled. Since her journey starts from home, we’ll start by marking this. And it might also be a good idea since we’re given the directions in terms of the compass directions to mark this on our diagram. We’ll use north representing up in our diagram.

Now, we’re going to want to start marking her journey. The first step in her journey is four miles east. The second part of her journey involves moving seven miles southeast. Next, we’re told that she walked six miles south. Then, after that, she walks five miles southwest. And then, finally, she walks three miles east. We want to determine the distance she would need to walk if she wanted to walk in a straight line home.

Now, there’s several different options we could use to find this distance. For example, we might want to do this geometrically by using trigonometry. However, by far, the easiest method is to put a coordinate system onto this diagram with the home point starting as the origin and then just find the coordinates where she ended up after her walk. Another option could be to represent each part of this journey as a vector. Then we could notice the sum of all of these vectors has to be equal to the zero vector because it starts and ends at the same point. Either of these methods will work. It’s personal preference which one you would want to use.

We’ll do the method involving vectors. However, it’s not necessarily the easiest method in this case. So let’s start by finding all of the vectors. Let’s start with the first part of her journey where she walks four miles east. Since we’ve chosen east to be our positive direction, the horizontal component of our vector will be four. And of course there’s no vertical component. This is just the vector four, zero.

We now want to represent the second part of her journey as a vector. We might initially want to call this the vector 𝐚, 𝐛. However, we can notice something. Since she’s traveling southeast in this section, every unit she moves east, she will move one unit south. This means we can be more specific with our vector. If she moves 𝐚 units east, then she needs to move 𝐚 units south. The vector would be of the form 𝐚, negative 𝐚, where 𝐚 is some positive number.

But this isn’t the only thing we know about this vector. We also know that its magnitude is going to be equal to seven because she traveled seven miles in this section. We can use this to find the value of 𝐚. The magnitude of this vector will be the square root of the sum of the squares of its components. This is the square root of 𝐚 squared plus negative 𝐚 all squared, and this should be equal to seven.

Now, all we need to do is rearrange this equation for 𝐚. We square both sides, divide through by two, and then take the square root of both sides. Remember, we want the positive square root. We get 𝐚 is equal to seven over root two. And we can simplify this. We can rationalize our denominator. We multiply the numerator and denominator through by root two. We get that 𝐚 is seven root two over two. So the vector representing the second part of her journey is the vector seven root two over two, negative seven root two over two.

We’re now going to want to do the same with the third part of her journey. So let’s clear some space and find this. In the third part of her journey, she just moves six miles south. So there’s no horizontal component to this part of the journey. And she moves six miles south. And we chose south to be the negative direction, so the vertical change is going to be negative six. So the vector representing the third part of her journey is the vector zero, negative six.

We’re then going to want to do the same for the fourth part of this journey. In this part of the journey, she walks five miles southwest. And moving southwest means she moves equal parts south and west. And both of these are in the negative directions. So our vector should be in the form negative 𝐚, negative 𝐚. So both components of this vector will be equal and negative.

And once again, we can find the value of 𝐚 by remembering the magnitude of this vector needs to be the distance walked in this section. It needs to be equal to five. And the working out here is very similar to the working out it was for the second part of our journey. We use our definition of the magnitude, square both sides, rearrange for our value of 𝐚, and remember that our value of 𝐚 should be positive. We get that 𝐚 should be equal to five root two over two. Therefore, the vector representing the fourth part of her journey is the vector negative five root two over two, negative five root two over two.

Let’s clear some space and then find the vector representing the fifth part of her journey. In the fifth part of this journey, she walks three miles east. So the horizontal component will be three, and the vertical component will be zero.

And now let’s say if she walked in a straight line home, we’ll call this part of the journey the vector 𝐯. Of course, the magnitude of our vector 𝐯 will be the answer to the question that we’re looking for. It will be the distance she needs to walk home if she walks home in a straight line. And one way of finding the magnitude of vector 𝐯 will be to find the components of 𝐯 and then use our formula for the magnitude. And one way of finding this is to notice something interesting about our diagram.

The initial point and terminal points of each of our vectors line up. And if we travel along all of our vectors, we see we end up back where we started. And remember, in this manner, traveling along vectors is the same as adding them together. So, in other words, the sum of these six vectors is equal to zero. We can use this to find the components of our vector 𝐯.

First, we’re going to need to add together all of the vectors in this diagram except our vector 𝐯. Adding all of these vectors together gives us the following expression. And to add these together, all we do is add their corresponding components together. And if we do this and simplify, we get the vector seven plus root two, negative six minus six root two. And if we add this vector to our vector 𝐯, then we should get the zero vector. And the only way this can happen is if our vector 𝐯 has opposite signs for each of the components of this vector. So our vector 𝐯 is the vector negative seven minus root two, six plus six root two.

Now, all we need to do is remember that the magnitude of our vector 𝐯 is going to be the distance that she needs to walk home if she walks home in a straight line. And we can find the magnitude of 𝐯 now that we have the components of 𝐯. It’s equal to the square root of the sum of the squares of its components. The magnitude of 𝐯 is the square root of negative seven minus root two all squared plus six plus six root two all squared. And if we calculate this expression, to three decimal places, we get 16.752. And remember, this represents a distance. And we know we’ve been working in miles, so we’ll give this the unit of miles.

In this question, we were able to turn a real-world problem about a woman walking home into a problem involving vectors. We were then able to use what we knew about vectors to find the total distance she was from home. We were able to show that she’d walked 25 miles. And if she were to walk home in a straight line, she would need to walk approximately 16.752 miles.

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