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The problem involves determining the moment of a force **F** about the **Oa** axis, which is in 3D space. The force vector is provided, and we have a hint that involves using the cross product and dot product to find the projection.

### Given:
- **Force vector**: \(\mathbf{F} = 50\hat{i} - 20\hat{j} + 20\hat{k} \, \text{N}\)
- Coordinates of point O (origin) and A (for the axis projection):
  - Position vector of point **a** relative to **O** can be written as \(\mathbf{r_{Oa}} = 2\hat{i} + 1\hat{j} + 3\hat{k} \, \text{m}\) (from the diagram).
  - Coordinates of the point where the force is applied are given by the relative position: \(\mathbf{r_{OF}} = 2\hat{i} + 1\hat{j} + 6\hat{k} \, \text{m}\).

### To determine:
The moment of force **F** about the **Oa** axis.

### Step 1: Moment about point O
The moment **M** about point O is calculated using the cross product of the position vector \(\mathbf{r}\) from O to the point of application of the force and the force vector \(\mathbf{F}\):
\[
\mathbf{M_O} = \mathbf{r_{OF}} \times \mathbf{F}
\]
Here,
\[
\mathbf{r_{OF}} = 2\hat{i} + 1\hat{j} + 6\hat{k}
\]
and
\[
\mathbf{F} = 50\hat{i} - 20\hat{j} + 20\hat{k}
\]

### Step 2: Project the moment onto the Oa axis
The moment about the Oa axis is the projection of **M_O** onto the unit vector along **Oa**:
\[
\mathbf{M_{Oa}} = \left( \mathbf{M_O} \cdot \hat{u}_{Oa} \right) \hat{u}_{Oa}
\]
where \(\hat{u}_{Oa}\) is the unit vector along the Oa axis, and it can be determined from the vector \(\mathbf{r_{Oa}}\).

Let me calculate this step by step.### Solution:

1. **Moment about point O**: The moment vector **M_O** is:
   \[
   \mathbf{M_O} = 140\hat{i} + 260\hat{j} - 90\hat{k} \, \text{Nm}
   \]

2. **Moment about the Oa axis** (projection of the moment onto the Oa axis) is:
   \[
   \mathbf{M_{Oa}} = 38.57\hat{i} + 19.29\hat{j} + 57.86\hat{k} \, \text{Nm}
   \]

This gives the moment of force **F** about the **Oa** axis in Cartesian vector form.

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Would you like further details on any of these steps or any related concepts?

Here are 5 related questions for deeper understanding:
1. How do you compute the cross product between two vectors?
2. What is the significance of the dot product when projecting vectors?
3. Why is it important to use a unit vector in projection calculations?
4. Can the magnitude of a moment be calculated from its vector components?
5. How do we interpret the physical meaning of the moment in 3D?

**Tip**: Always verify that vectors are properly aligned with the right-hand rule when calculating moments using the cross product.

Determine the moment of the force F about the Oa axis. Express the results as a Cartesian vector. Given F = (50i - 20j + 20k) N and coordinates of point O (origin) and a (for the axis projection).

Learn how to determine the moment of a force F = (50i - 20j + 20k) N about the Oa axis using cross and dot product techniques. This detailed solution covers vector calculus concepts and is suitable for engineering students.

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