R= the square root of x^2 + y^2

[CarlosNunes]

Homework Statement

This problem is #17 in Chapter 3 of Giancoli's 5th Edition Physics.

The summit of a mountain, 2085 m above base camp, is measured on a map to be 4580 m horizontally from the camp in a direction 32.4 degrees west of north, What are the x, y, and z components of the displacement vector from camp to summit? What is the length? Choose the x-axis east, y-axis north, and z axis up.

Homework Equations

R= the square root of x^2 + y^2

The Attempt at a Solution

 

[Cephalopoda]

The angle refers to that between the z and x axis. The hypotenuse of the triangle made between these two is 4580. You then have the means of calculating the x and z displacements. Sin(32.4) X 4580 is the x displacement. Cos(32.4) X 4580 is the z displacement. Don't forget to make the x vector negative as you are traveling west, and positive is east. Intro physics is all about triangles and separating a magnitude (hypotenuse) into its vector components.

 

[m2003]

To solve this problem, we first need to understand the given information. The problem is asking for the displacement vector from the base camp to the summit of a mountain. The displacement vector is a vector that represents the change in position from one point to another. In this case, the x and y components of the displacement vector represent the horizontal distance from the base camp to the summit, while the z component represents the vertical distance.

To find the x and y components, we can use trigonometry. We know that the horizontal distance from the base camp to the summit is 4580 m, and the angle between the direction of the summit and the north direction is 32.4 degrees. Using the sine and cosine functions, we can determine that the x component is 4580*cos(32.4) = 3854.9 m and the y component is 4580*sin(32.4) = 2436.6 m.

To find the z component, we can use the given information that the summit is 2085 m above the base camp. Therefore, the z component is simply 2085 m.

To find the length of the displacement vector, we can use the Pythagorean theorem. The length of the displacement vector, R, is equal to the square root of the sum of the squares of its components. In this case, R = √(3854.9^2 + 2436.6^2 + 2085^2) = 5556.9 m.

In conclusion, the x, y, and z components of the displacement vector from the base camp to the summit are 3854.9 m, 2436.6 m, and 2085 m, respectively. The length of the displacement vector is 5556.9 m.