**Vector Force and Axis Angles Calculation***Problem Statement:*1/4 A force is specified by the vector **F** = 160i + 80j - 120k N. Calculate the angles made by **F** with the positive x-, y-, and z-axes.*Detailed Explanation:*Given the force vector **F** = 160i + 80j - 120k N, we will calculate the angles that this force vector makes with each of the positive x-, y-, and z-axes.1. **Force Vector Components:** - **Fx** (Component along the x-axis) = 160 N - **Fy** (Component along the y-axis) = 80 N - **Fz** (Component along the z-axis) = -120 N2. **Magnitude of the Force Vector (|**F**|):** The magnitude of vector **F** can be calculated using the Pythagorean theorem for three dimensions: \[ |F| = \sqrt{(Fx)^2 + (Fy)^2 + (Fz)^2} \] Substituting the given values: \[ |F| = \sqrt{(160)^2 + (80)^2 + (-120)^2} \] \[ |F| = \sqrt{25600 + 6400 + 14400} \] \[ |F| = \sqrt{46400} \] \[ |F| \approx 215.4 \text{ N} \]3. **Angles with the Axes:** - **Angle with the x-axis (α):** This angle can be found using the dot product definition for vectors: \[ \cos(α) = \frac{Fx}{|F|} \] Substituting the values: \[ \cos(α) = \frac{160}{215.4} \] \[ α = \cos^{-1}\left(\frac{160}{215.4}\right) \] \[ α \approx \cos^{-1}(0.743) \] \[ α \approx 42.1° \] - **Angle with the y-axis (β):

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Answered: 1/4 A force is specified by the vector…

1/4 A force is specified by the vector F = 160i+ 80j - 120k N. Calculate the angles made by F with the positive x-, y-, and z-axes.

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter5: Trigonometric Functions: Right Triangle Approach

Section5.1: Angle Measure

Problem 4E: Object A is travelling along a circle of radius 2, and Object B is travelling along a circle of...

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Question

**Vector Force and Axis Angles Calculation**

*Problem Statement:*

1/4 A force is specified by the vector **F** = 160i + 80j - 120k N. Calculate the angles made by **F** with the positive x-, y-, and z-axes.

*Detailed Explanation:*

Given the force vector **F** = 160i + 80j - 120k N, we will calculate the angles that this force vector makes with each of the positive x-, y-, and z-axes.

1. **Force Vector Components:**
- **Fx** (Component along the x-axis) = 160 N
- **Fy** (Component along the y-axis) = 80 N
- **Fz** (Component along the z-axis) = -120 N

2. **Magnitude of the Force Vector (|**F**|):**

The magnitude of vector **F** can be calculated using the Pythagorean theorem for three dimensions:

\[
|F| = \sqrt{(Fx)^2 + (Fy)^2 + (Fz)^2}
\]

Substituting the given values:

\[
|F| = \sqrt{(160)^2 + (80)^2 + (-120)^2}
\]
\[
|F| = \sqrt{25600 + 6400 + 14400}
\]
\[
|F| = \sqrt{46400}
\]
\[
|F| \approx 215.4 \text{ N}
\]

3. **Angles with the Axes:**
- **Angle with the x-axis (α):**

This angle can be found using the dot product definition for vectors:
\[
\cos(α) = \frac{Fx}{|F|}
\]
Substituting the values:
\[
\cos(α) = \frac{160}{215.4}
\]
\[
α = \cos^{-1}\left(\frac{160}{215.4}\right)
\]
\[
α \approx \cos^{-1}(0.743)
\]
\[
α \approx 42.1°
\]

- **Angle with the y-axis (β):

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