A woman starts walking from home and walks 4 miles east, 7...

Name: Problem 558, Chapter 8: Precalculus
Uploaded: 2019-07-27T03:04:31-08:00
Duration: 6 min 28 s
Channel: Yifan Xu
Description: A woman starts walking from home and walks 4 miles east, 7 miles southeast, 6 miles south, 5 miles southwest, and 3 miles east. How far has she walked? If she walked straight home, how far would she have to walk?

A woman starts walking from home and walks 4 miles east, 7 miles southeast, 6 miles south, 5 miles southwest, and 3 miles east. How far has she walked? If she walked straight home, how far would she have to walk?

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is used to calculate the magnitude of the resultant displacement vector by relating the horizontal and vertical components. It provides a method to compute the direct distance between two points in a coordinate system.

Vector Decomposition

Vector decomposition is the process of breaking vectors into their x and y components using trigonometric functions. This allows complex directional movements, such as those at an angle, to be handled with simpler arithmetic, and is fundamental when applying vector addition.

Vector Addition

Vector addition involves summing multiple directional movements by combining their horizontal (x) and vertical (y) components. It is essential for determining the net displacement when several movements in different directions are involved.

Displacement vs. Total Distance

Total distance is the sum of all lengths of the path taken, irrespective of direction, while displacement is the straight-line distance from the starting point to the ending point. Understanding the difference between these two is crucial when analyzing motion.

Transcript

00:01 Alright, so a woman starts working from home and works 4 miles east, 7 miles south east, 6 miles south, 5 miles southwest, and 3 miles east.

00:12 So let me just visualize everything.

00:22 All right, so a woman firstly works 4 miles east, which is this direction to here.

00:31 So this is 4.

00:33 And then 7 miles southeast.

00:37 Which is that direction.

00:42 And then six miles south, which is here, and then five miles south -west, which is here.

00:54 And finally, three miles east, which is here.

01:06 All right, so we have, this is four miles, and this is seven miles, and this is six miles, this is five miles, and finally, this is three miles.

01:19 And this is the final destination.

01:24 This is the initial starting point.

01:29 And the first question is, how far has she traveled? it's very simple, right? because you just add up every steps together.

01:36 So it's 4 plus 7 plus 6 plus 5 plus 3, which is 25 miles.

01:49 And the second question is a bit trickier.

01:52 The question is to calculate how far will she have to travel if she worked straight home, which means that from where it is right now, straight back home, i want to know the distance.

02:14 All right, so the key thing for this question is in order to find the distance, we want to find the x and y component respectively for this distance.

02:29 So for y component and for x component, we can respectively draw a line to help us visualize everything.

02:49 So we know that this is the y component, and this is the x component.

03:01 Let me just illustrate the x component first...

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