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Let's analyze the questions using Principal Component Analysis (PCA) to determine the correct directions of projection.

### Question 3:
The curve in **Figure 1** appears to be an ellipse with its major axis aligned along the \(x_2\) (vertical) direction.

- **PCA Explanation**: 
  - PCA projects data points onto directions of maximum variance. The direction with the highest variance is the principal component.
  - For an ellipse, the direction of the major axis (the longer axis) is the first principal component, as it contains the most variance in the data.

- **Answer**: Since the major axis of the ellipse is along the \(x_2\) axis, the direction of projection should be along the **Y-axis**.
  - **Correct Answer**: **b) Y-axis**

### Question 4:
The curve in **Figure 2** shows an ellipse rotated counterclockwise.

- **PCA Explanation**:
  - Here, the major axis of the ellipse is not aligned with the coordinate axes but at some angle.
  - The direction of maximum variance (the principal component) will be along the major axis of this rotated ellipse.

- **Options**:
  - **45°** with the x-axis in an anticlockwise direction corresponds to the diagonal line from the origin up to the right (along \(y = x\)).
  - **135°** with the x-axis in an anticlockwise direction corresponds to the diagonal line from the origin down to the left (along \(y = -x\)).

- **Answer**: The major axis of the ellipse in Figure 2 is likely aligned with **45°** relative to the x-axis.
  - **Correct Answer**: **c) Direction making an angle of 45° with x-axis in anticlockwise direction.**

Would you like further details or have any questions?

### Related Questions:
1. How do you determine the direction of the principal component using covariance matrices?
2. Can PCA be used on datasets with more than two dimensions, and if so, how?
3. How does PCA handle noise in the data?
4. What happens if the data points are uniformly distributed across all directions?
5. How would the PCA projection change if the data distribution were non-linear, like in a spiral?

### Tip:
Always standardize your data before applying PCA, especially if the features have different units or scales.

Analyze the correct direction of projection using Principal Component Analysis (PCA) for data points on elliptical curves. Learn about the relationship between PCA, the covariance matrix, and the projection direction, focusing on identifying the principal component for different elliptical orientations.

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