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The problem involves finding the magnitude and the coordinate direction angles of a force vector \( \mathbf{F} \) acting on a bracket. The given components of the force are \( F_x = 300 \, \text{N} \), \( F_z = 600 \, \text{N} \), and the angle \( \beta = 60^\circ \) between \( F \) and the \( x \)-axis. We also need to find the direction angles \( \alpha \) and \( \gamma \).

### Step 1: Find the Magnitude of the Force \( F \)
Since \( \beta \) is the angle with the \( x \)-axis, the magnitude of \( F \) is related to \( F_x \) by the cosine of \( \beta \):
\[ F = \frac{F_x}{\cos \beta} \]
\[ F = \frac{300}{\cos 60^\circ} = \frac{300}{0.5} = 600 \, \text{N} \]

### Step 2: Find \( F_y \)
Using the Pythagorean theorem on the components of \( \mathbf{F} \), we have:
\[ F = \sqrt{F_x^2 + F_y^2 + F_z^2} \]
Plugging in the known values and solving for \( F_y \):
\[ 600 = \sqrt{300^2 + F_y^2 + 600^2} \]
\[ 360000 = 90000 + F_y^2 + 360000 \]
\[ F_y^2 = 90000 \]
\[ F_y = 300 \, \text{N} \]

### Step 3: Find the Coordinate Direction Angles \( \alpha \), \( \beta \), and \( \gamma \)
The coordinate direction angles can be found using:
\[ \cos \alpha = \frac{F_x}{F} \]
\[ \cos \beta = \frac{F_y}{F} \]
\[ \cos \gamma = \frac{F_z}{F} \]

\[ \alpha = \cos^{-1} \left( \frac{300}{600} \right) = \cos^{-1}(0.5) = 60^\circ \]
\[ \beta = \cos^{-1} \left( \frac{300}{600} \right) = \cos^{-1}(0.5) = 60^\circ \]
\[ \gamma = \cos^{-1} \left( \frac{600}{600} \right) = \cos^{-1}(1) = 0^\circ \]

Hence, the magnitude of \( \mathbf{F} \) is \( 600 \, \text{N} \), and the coordinate direction angles are \( \alpha = 60^\circ \), \( \beta = 60^\circ \), and \( \gamma = 0^\circ \).

Do you need further details or have any questions about these calculations?

Here are some related questions:
1. What would the direction angles be if \( F_x \) or \( F_z \) were different?
2. How would a negative component affect the direction angles?
3. What is the significance of each direction angle in vector components?
4. Can you explain how to resolve a vector into its components in three-dimensional space?
5. How does changing the angle \( \beta \) affect the magnitude and components of \( \mathbf{F} \)?

**Tip:** To better understand vector operations, visualize them using vector component diagrams. This can help with comprehending the geometric relationships between the angles and the vector components.

The force F acts on the bracket within the octant shown. If the magnitudes of the x and z components of F are Fx = 300 N and Fz = 600 N, respectively, and β = 60°, determine the magnitude of F and its y-component. Also, find the coordinate direction angles α and γ.

This problem demonstrates how to find the magnitude of a force vector acting on a bracket, given its x and z components, as well as the y-component. It involves key concepts such as vector decomposition, the Pythagorean Theorem, and trigonometric relations. Suitable for university-level engineering or physics students.

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