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To determine the relationship between the given planes, we'll compare their normal vectors. The normal vector of a plane given by the equation \( ax + by + cz = d \) is \( \mathbf{n} = \langle a, b, c \rangle \).

### Step 1: Identify the normal vectors
1. **Reference Plane:** \( -5x - 4y - 3z = 1 \)  
   Normal vector: \( \mathbf{n_1} = \langle -5, -4, -3 \rangle \)

2. **Plane 1:** \( 4x - 5y = 4 \)  
   This equation can be rewritten as \( 4x - 5y + 0z = 4 \).  
   Normal vector: \( \mathbf{n_2} = \langle 4, -5, 0 \rangle \)

3. **Plane 2:** \( 25x + 20y + 15z = 2 \)  
   Normal vector: \( \mathbf{n_3} = \langle 25, 20, 15 \rangle \)

4. **Plane 3:** \( -5x - 3y - 3z = -3 \)  
   Normal vector: \( \mathbf{n_4} = \langle -5, -3, -3 \rangle \)

### Step 2: Compare the normal vectors

- **Parallel Planes:** The normal vectors are proportional (i.e., one is a scalar multiple of the other).
- **Orthogonal Planes:** The dot product of the normal vectors is zero.
- **Neither:** If the planes are neither parallel nor orthogonal.

#### Comparison:

1. **Plane 1 vs. Reference Plane:**
   - \( \mathbf{n_1} = \langle -5, -4, -3 \rangle \)
   - \( \mathbf{n_2} = \langle 4, -5, 0 \rangle \)
   - Dot product: \( (-5)(4) + (-4)(-5) + (-3)(0) = -20 + 20 + 0 = 0 \)  
   The planes are **orthogonal**.

2. **Plane 2 vs. Reference Plane:**
   - \( \mathbf{n_3} = \langle 25, 20, 15 \rangle \)
   - The vector \( \mathbf{n_3} \) is proportional to \( \mathbf{n_1} \) because \( 25 = 5(-5) \), \( 20 = 5(-4) \), and \( 15 = 5(-3) \).
   The planes are **parallel**.

3. **Plane 3 vs. Reference Plane:**
   - \( \mathbf{n_4} = \langle -5, -3, -3 \rangle \)
   - The vectors \( \mathbf{n_1} \) and \( \mathbf{n_4} \) are not proportional, and their dot product is not zero.
   The planes are **neither**.

### Final Answers:
- **Plane 1:** Orthogonal (A)
- **Plane 2:** Parallel (B)
- **Plane 3:** Neither (C)

Would you like to explore this further or have any questions about the steps?

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**Related Questions:**

1. How do you find the angle between two planes?
2. What is the geometric interpretation of the dot product?
3. Can two non-parallel planes in 3D space ever be orthogonal?
4. How do you determine if two planes are identical?
5. What are the conditions for two planes to intersect in a line?

**Tip:** Always verify the consistency of signs when checking if vectors are proportional, as small mistakes can lead to incorrect conclusions.

Compare the planes below to the plane -5x - 4y - 3z = 1. Match the letter corresponding to the words parallel, orthogonal, or 'neither,' which describes the relation of the two planes: 1) 4x - 5y = 4, 2) 25x + 20y + 15z = 2, 3) -5x - 3y - 3z = -3.

Explore the relationships between planes based on their normal vectors. Learn how to identify if planes are parallel, orthogonal, or neither using vector algebra and dot products. This problem is ideal for students in Grades 10-12.

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