Solve the following problem using analytical techniques:...

Solve the following problem using analytical techniques: Suppose you walk $18.0 \mathrm{~m}$ straight west and then $25.0 \mathrm{~m}$ straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements $\mathbf{A}$ and $\mathbf{B}$, as in Figure 3.55, then this problem asks you to find their sum $\mathbf{R}=\mathbf{A}+\mathbf{B}$. Figure 3.55 (FIGURE CAN'T COPY)

Key Concepts

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. It is fundamental in physics and engineering when dealing with quantities that have both magnitude and direction. This method involves adding corresponding components of vectors, often breaking them down into perpendicular directions such as x and y axes, and then combining them to obtain the total effect of multiple vector displacements.

Displacement

Displacement refers to the change in position of an object and is a vector quantity, meaning it has both magnitude and direction. Unlike distance, displacement is concerned with the shortest path between an object’s initial and final positions. It is crucial when analyzing movements in space, as it represents the net change in location regardless of the actual path taken.

Pythagorean Theorem

The Pythagorean theorem is a principle in geometry used to relate the sides of a right triangle. When two vectors are perpendicular, their resultant magnitude can be found using the theorem by taking the square root of the sum of the squares of the components. This is a key analytical technique for determining the magnitude of the resultant displacement when the component vectors are at right angles.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are used to relate the angles of a triangle to its side lengths. In the context of vector addition, these functions are essential for determining the direction of the resultant vector, often by calculating angles with respect to a reference axis using inverse trigonometric functions.

Compass Directions

Compass directions provide a way to describe the orientation of a vector in terms of the cardinal points—north, south, east, and west. After finding the magnitude and the angle of the resultant displacement, translating the angle into a compass bearing allows for understanding and communicating the direction in a practical, real-world context.

Transcript

00:01 For this example, we're going to figure out the magnitude and direction of our displacement vector after having traveled 18 meters to the west and then 25 meters to the north.

00:11 And we're going to be doing this strictly just with analytical methods, which means just vector algebra.

00:20 So really what this problem is going to boil down to is figuring out what our resultant vector is.

00:30 When we know that our resultant r is just going to be a sum of two vectors.

00:38 The first is the vector that we follow along when traveling west, and the second is the one we follow along when traveling north.

00:48 So our resultant should look something like this.

00:53 And we can call this second one b and this first one a.

00:58 So we know that r is just going to be the sum of a and b.

01:05 And what this means is that the components of r, namely rx and ry, are just going to be the sum of the individual components of a and b.

01:18 So r sub x is going to be a sub x plus b sub x, and r sub y is going to be a sub y plus b sub y.

01:30 So let's break down these components for a and b, and then we'll be able to find our components of r.

01:36 So for a, we know that it is strictly in the horizontal direction, so it should only have an x component.

01:50 So right off the bat, we can say that the y component is zero, and the x component we know is just going to be 18 meters, because that's the distance we traveled west.

02:04 Similarly, for b, we know there's going to be no x component because the path is strictly vertical...

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