Chapter 3, Problem 15P

To determine

1. The components of displacement vector from camp to summit.
2. The magnitude.

Answer to Problem 15P

Solution:

1. The components of the displacement vector $\overrightarrow{d}$ are: The x -component is -2845 m, the y -component is 3589 m and the z-component is2450 m.
2. The magnitude $\left|\overrightarrow{d}\right|$ of the displacement vector $\overrightarrow{d}$ is 5194 m.

Explanation of Solution

Explanation:Given:The z-component $d_z$ of the displacement vector $\overrightarrow{d}$ is 2450 m and the horizontal distance from base camp to base summit is 4580 m with a direction of 38.4 °.

Formula used:

The situation can be shown below:

  

The components of the displacement vector $\overrightarrow{d}$ can be obtained using:

  $d_x=4580\cos \theta \mathrm{...}\left(1\right)$

And

  $d_y=4580\sin \theta \mathrm{...}\left(2\right)$

Where:

• $d_x$ is the x-component of the displacement vector $\overrightarrow{d}$ .
• $d_y$ is the y-component of the displacement vector $\overrightarrow{d}$ .
• $\theta$ is the angle measured from the + x -axis.

In this case, the z-component is known.

The magnitude $\left|\overrightarrow{d}\right|$ of the displacement vector $\overrightarrow{d}$ is defined as:

  $\left|\overrightarrow{d}\right|=\sqrt{d_x{}^2+d_y{}^2+d_z{}^2}\mathrm{...}\left(3\right)$

Where:

• $d_x$ is the x-component of the displacement vector $\overrightarrow{d}$ .
• $d_y$ is the y-component of the displacement vector $\overrightarrow{d}$ .
• $d_z$ is the z-component of the displacement vector $\overrightarrow{d}$ .

Calculation: Substitute values into equation (1) and solve:

  $d_x=4580\cos \left({90}^o+{38.4}^o\right)=-2845\text{m}$

Substitute values into equation (2) and solve:

  $d_y=4580\sin \left({90}^o+{38.4}^o\right)=3589\text{m}$

Substitute values in equation (3) and solve:

  $\begin{array}{l}\left|\overrightarrow{d}\right|=\sqrt{{\left(-2845\right)}^2+{\left(3589\right)}^2+{\left(2450\right)}^2}\\ \left|\overrightarrow{d}\right|=5194\text{m}\end{array}$

Therefore, the magnitude $\left|\overrightarrow{d}\right|$ of the displacement vector $\overrightarrow{d}$ is 5194 m.

Conclusion:

The components of the displacement vector $\overrightarrow{d}$ are: The x-component is -2845 m, the y-component is 3589 m and the z-component is2450 m, and the magnitude $\left|\overrightarrow{d}\right|$ of the displacement vector $\overrightarrow{d}$ is 5194 m.