A woman starts walking from home and walks 6 miles at 40^∘ north of east, then 2 miles at 15^∘ e

A woman starts walking from home and walks 6 miles at 40^∘...

Key Concepts

Vector Addition

This concept involves combining multiple displacement vectors to obtain a single resultant vector that represents the overall change in position. It is fundamental in determining the net effect of several movements in different directions.

Component Decomposition

This key concept refers to breaking down each vector into its horizontal and vertical components using trigonometric functions. By resolving vectors into perpendicular components, one can simplify the process of adding them together.

Coordinate System Orientation

Understanding how to assign directions such as north, east, south, and west to a coordinate system is essential. This allows for accurate representation of directional movement as vectors in a two-dimensional plane.

Trigonometric Functions in Vector Analysis

Trigonometric functions like sine and cosine are used to calculate the magnitude of a vector’s components based on its direction. They are vital for converting between a vector’s magnitude and its directional components.

Resultant Magnitude and Direction Calculation

After summing the individual components of all vectors, the overall displacement is found by computing the magnitude using the Pythagorean theorem and determining the direction with inverse trigonometric functions. This process provides both the distance and the bearing for the resultant vector.

Transcript

00:01 All right, so for this question, a woman starts working from home and works six miles at 40 degree north of east, and then two miles at 15 east of south, and then five miles at 30 degrees south of west.

00:14 So let me just visualize the process.

00:23 All right, let's say the origin point, the zero zero point is the home.

00:28 So she firstly starts to walk six miles at 40 degree north of east, which is going to be, this position.

00:39 So this is the destination when he walks six miles and this is 40 degree.

00:48 And then she walks two miles at 15 east of south, which is going to be this direction.

01:01 So this is 15 degree.

01:08 And then she walks five miles at 30 south of west, which is going to be let's say 30 is this direction so this is the final position all right so this graph is a very so this is a very rough graph so the position for each of this point and a relative position is unknown i just draw the scratch i just draw the scratch graph so it's not particularly accurate so the question is to so the question is if she walked straight home how far would she have to travel and in what direction? so in essence, the question is to ask, what is the distance between the starting point and the ending point? so what is the distance of this red line and what is the angle? so let's say, in order to get the red line, we want to get the x component, a y component of this red line.

02:26 So let's first decompose this into x.

02:32 X and y's.

02:33 So this is y and this is x, which represents the magnitude.

02:43 So the y here is actually positive.

02:45 We're just scared about the magnitude.

02:48 All right, so in order to get the x here, we just used the three information and the angles.

02:58 So firstly, we're going to get this distance, which is going to be six times code so 6 times cosine 40 because this is 40 degree or this is 40 degree so cosine 40 times 6 is this distance so we have that we want to calculate this distance which is going to be two times sine 15 degree all right after we get that we want to subtract this distance we want to subtract this which is going to be 3 times sine 30 all right so this is the distance of x which when you put everything in a calculator you're going to get the answer being 3 .61 so that's the distance of x so now the question is to calculate the distance why so for y we first we want to calculate the total distance here, which composes two parts...

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