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?

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.