Find the condition that the angles 𝜃𝑥 and 𝜃𝑦 must satisfy in order for them to represent direction angles: 
a) If 𝜃𝑥 ≥ 90 and 𝜃𝑦 ≤ 90 
b) If 𝜃𝑥 ≥ 90 and 𝜃𝑦 ≥ 90 

To solve this, we need to understand the geometric interpretation of **direction angles** for a vector. In 2D, the **direction angles** \(\theta_x\) and \(\theta_y\) correspond to the angles between the vector and the positive \(x\)-axis and \(y\)-axis, respectively.

For direction angles to be valid, they must satisfy specific conditions based on the quadrant they are located in. Let's explore each case in detail.

### a) If \(\theta_x \geq 90^\circ\) and \(\theta_y \leq 90^\circ\):
- \(\theta_x \geq 90^\circ\): This indicates that the vector is in either the 2nd or 3rd quadrant since the angle with the \(x\)-axis is more than 90°.
- \(\theta_y \leq 90^\circ\): This indicates that the angle with the \(y\)-axis is less than or equal to 90°, so the vector lies in the 1st or 2nd quadrant.

Thus, **both conditions can only hold true in the 2nd quadrant**, where:
- \(\theta_x\) is between 90° and 180°.
- \(\theta_y\) is between 0° and 90°.

### b) If \(\theta_x \geq 90^\circ\) and \(\theta_y \geq 90^\circ\):
- \(\theta_x \geq 90^\circ\): The vector is in either the 2nd or 3rd quadrant.
- \(\theta_y \geq 90^\circ\): The vector is in either the 3rd or 4th quadrant.

Thus, **both conditions can only hold true in the 3rd quadrant**, where:
- \(\theta_x\) is between 90° and 180°.
- \(\theta_y\) is between 90° and 180°.

### Summary of Conditions:
- For \(\theta_x \geq 90^\circ\) and \(\theta_y \leq 90^\circ\), the vector must be in the **2nd quadrant**.
- For \(\theta_x \geq 90^\circ\) and \(\theta_y \geq 90^\circ\), the vector must be in the **3rd quadrant**.

### Do you need any further details or have questions?

Here are 5 relative questions you might consider:
1. How are direction angles different in 3D compared to 2D?
2. What are the conditions for a vector to lie in the 1st or 4th quadrant?
3. Can direction angles be negative, and how would that change these conditions?
4. How can you use direction cosines to represent vectors in 2D and 3D?
5. What is the relationship between direction angles and polar coordinates?

**Tip:** Direction angles are closely related to the concept of vector components and trigonometric functions (sine, cosine). Understanding these angles helps in visualizing the direction of vectors in both 2D and 3D spaces.

Find the condition that the angles 𝜃𝑥 and 𝜃𝑦 must satisfy in order for them to represent direction angles: a) If 𝜃𝑥 ≥ 90 and 𝜃𝑦 ≤ 90 b) If 𝜃𝑥 ≥ 90 and 𝜃𝑦 ≥ 90

Explore the conditions that angles θx and θy must satisfy to represent direction angles in 2D. This detailed explanation covers how these angles define vector direction, with cases where θx ≥ 90° and θy ≤ 90° (2nd quadrant) and θx ≥ 90° and θy ≥ 90° (3rd quadrant). Ideal for high school students studying trigonometry and vector analysis.

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