https://file.aiquickdraw.com/mathbot/file/1723353912629wp0r3uny.jpg

### Problem Description:

The problem is asking for the equations of planes given a point on the plane and the normal vector to the plane.

### Problem Setup:

For each sub-part:

- **Equation of Plane**: \( ax + by + cz = d \)
- **Given**:
  - **Point** \((x_1, y_1, z_1)\)
  - **Normal Vector** \( \mathbf{n} = ai + bj + ck \)
- **Plane Equation in Normal Form**:
\[ a(x - x_1) + b(y - y_1) + c(z - z_1) = 0 \]

### (i) Sub-part
- **Point**: \((3, 1, 2)\)
- **Normal Vector**: \( \mathbf{n} = 2i - 3j + 4k \)

Using the normal form:
\[
2(x - 3) - 3(y - 1) + 4(z - 2) = 0
\]
Expanding:
\[
2x - 6 - 3y + 3 + 4z - 8 = 0
\]
Simplifying:
\[
2x - 3y + 4z = 11
\]

### (ii) Sub-part
- **Point**: \((-2, 5, 3)\)
- **Normal Vector**: \( \mathbf{n} = 3i - j + 2k \)

Using the normal form:
\[
3(x + 2) - (y - 5) + 2(z - 3) = 0
\]
Expanding:
\[
3x + 6 - y + 5 + 2z - 6 = 0
\]
Simplifying:
\[
3x - y + 2z = -5
\]

### (iii) Sub-part
- **Point**: \((4, 0, 1)\)
- **Normal Vector**: \( \mathbf{n} = 2i + j - k \)

Using the normal form:
\[
2(x - 4) + (y - 0) - (z - 1) = 0
\]
Expanding:
\[
2x - 8 + y - z + 1 = 0
\]
Simplifying:
\[
2x + y - z = 7
\]

### Final Answer:
1. \(2x - 3y + 4z = 11\)
2. \(3x - y + 2z = -5\)
3. \(2x + y - z = 7\)

Would you like further details on these solutions or help with any specific part of the process?

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### Related Questions
1. What are the steps to derive the plane's equation from a given point and normal vector?
2. How can we verify if a point lies on a given plane?
3. What is the significance of the normal vector in defining a plane?
4. How can the distance from a point to a plane be calculated?
5. What method can be used to find the intersection line of two planes?
6. How do changes in the normal vector affect the orientation of the plane?
7. Can the equation of a plane be written in parametric form?
8. How to find the equation of a plane passing through three non-collinear points?

### Tip
Always double-check the arithmetic when expanding and simplifying the equation of a plane. Small mistakes can lead to incorrect final equations.

Explore examples of finding the equation of planes using a given point and normal vector. Includes step-by-step solutions and calculations for each example. Suitable for advanced high school students and anyone studying geometry and vectors.

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