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To determine the angle \( \theta \) between the wire \( AB \) and the y-axis of the pole, we can apply the dot product formula. 

Given:
- \( \mathbf{r}_{AC} = -3j \, \text{ft} \) (which is the y-axis direction vector),
- \( \mathbf{r}_{AB} = (2 - 0)\hat{i} + (2 - 3)\hat{j} + (-2 - 0)\hat{k} = 2\hat{i} - 1\hat{j} - 2\hat{k} \, \text{ft} \).

To compute the angle, we use the dot product formula:

\[
\mathbf{r}_{AC} \cdot \mathbf{r}_{AB} = |\mathbf{r}_{AC}| |\mathbf{r}_{AB}| \cos \theta
\]

### Step 1: Calculate the magnitudes
- \( |\mathbf{r}_{AC}| = \sqrt{(-3)^2} = 3 \, \text{ft} \),
- \( |\mathbf{r}_{AB}| = \sqrt{2^2 + (-1)^2 + (-2)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \, \text{ft} \).

### Step 2: Compute the dot product
The dot product of \( \mathbf{r}_{AC} \) and \( \mathbf{r}_{AB} \) is:

\[
\mathbf{r}_{AC} \cdot \mathbf{r}_{AB} = (0 \cdot 2) + (-3 \cdot -1) + (0 \cdot -2) = 0 + 3 + 0 = 3
\]

### Step 3: Solve for \( \theta \)
Using the dot product equation:

\[
3 = (3)(3) \cos \theta
\]

\[
\cos \theta = \frac{3}{9} = \frac{1}{3}
\]

Now, \( \theta \) is:

\[
\theta = \cos^{-1}\left( \frac{1}{3} \right) \approx 70.53^\circ
\]

### Final Answer:
The angle \( \theta \) between the y-axis and the wire \( AB \) is approximately \( 70.53^\circ \).

Would you like any further details or explanations?

Here are five related questions you might explore:
1. What is the general formula for the dot product of two vectors?
2. How does the magnitude of a vector relate to its components?
3. Can the dot product be used to determine if two vectors are perpendicular?
4. How does changing the direction of one vector affect the dot product?
5. In what real-world scenarios is finding the angle between vectors important?

**Tip:** When working with vectors in 3D, always make sure to use proper vector notation and understand how each component (i, j, k) affects the spatial orientation.

Determine the angle θ between the y-axis of the pole and the wire AB. You will want to use the dot-product approach to find the angle between the vectors r_AC = {-3j} ft and r_AB = {(2 - 0)i + (2 - 3)j (-2 - 0)k} ft.

Learn how to find the angle θ between the y-axis and the wire AB using the dot product of vectors. The problem involves calculating vector magnitudes, the dot product, and applying trigonometry. Suitable for engineering or physics students.

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