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I j k compnets for f1 f2 f3 find the resultant of f1f2f3 then find the alpha betta and gamma of The resultant 

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.

Explore how to calculate the resultant force vector and angles \(\alpha\), \(\beta\), and \(\gamma\) in 3D space. This problem involves analyzing forces \(\vec{F}_1\), \(\vec{F}_2\), and \(\vec{F}_3\) and determining their combined effect. Learn the step-by-step process, including vector addition and angle calculations. Suitable for advanced high school students and anyone interested in physics and vector analysis.

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