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.