Let's analyze the forces $\vec{F}_1$, $\vec{F}_2$, and $\vec{F}_3$ given in the image and find the resultant force vector, followed by calculating the angles $\alpha$, $\beta$, and $\gamma$ with the axes.

Step 1: Determine the Components of Each Force

Force $\vec{F}_1 = 50 \, \text{N}$

• Directed along the negative $x$-axis:
  • Components: $\vec{F}_1 = -50 \hat{i} + 0 \hat{j} + 0 \hat{k}$

Force $\vec{F}_2 = 150 \, \text{N}$

• Given at $60^\circ$ to the $x$-axis and $45^\circ$ to the $z$-axis in the $xy$-plane:
  • Components:
    • $F_{2x} = 150 \cos(60^\circ) = 75 \, \text{N}$
    • $F_{2y} = 150 \sin(60^\circ) \cos(45^\circ) = 91.92 \, \text{N}$
    • $F_{2z} = 150 \sin(60^\circ) \sin(45^\circ) = 91.92 \, \text{N}$

Force $\vec{F}_3 = 200 \, \text{N}$

• Directed along the positive $z$-axis:
  • Components: $\vec{F}_3 = 0 \hat{i} + 0 \hat{j} + 200 \hat{k}$

Step 2: Sum the Components to Find the Resultant Force

Sum the $i$, $j$, and $k$ components of all forces:

$\vec{F}_{\text{resultant}} = (F_{1x} + F_{2x} + F_{3x}) \hat{i} + (F_{1y} + F_{2y} + F_{3y}) \hat{j} + (F_{1z} + F_{2z} + F_{3z}) \hat{k}$

Now, let's plug in the values and calculate the resultant force and angles.Here are the results for the forces and their calculations:

Resultant Force Vector Components

$\vec{F}_{\text{resultant}} = 25.00 \hat{i} + 91.86 \hat{j} + 291.86 \hat{k} \, \text{N}$

Magnitude of the Resultant Force

$|\vec{F}_{\text{resultant}}| = 306.99 \, \text{N}$

Angles with the Axes

• Angle with $x$-axis ($\alpha$): $85.33^\circ$
• Angle with $y$-axis ($\beta$): $72.59^\circ$
• Angle with $z$-axis ($\gamma$): $18.07^\circ$

Would you like further details or have any questions?

Here are five related questions you might consider:

1. How would the resultant change if $\vec{F}_1$ was along a different axis?
2. What are the implications of a negative angle value in vector directions?
3. How does changing the angle of $\vec{F}_2$ affect the resultant?
4. How can you verify these calculations using vector projection methods?
5. What would be the effect of adding another force at a different angle?

Tip: Always break forces into components when dealing with vectors in 3D space to simplify your calculations.