To write the equation of a plane in 3-dimensional space, you generally use the following form:

\[
Ax + By + Cz = D
\]

where:
- \( (x, y, z) \) are the coordinates of any point on the plane,
- \( A, B, C \) are the coefficients that represent the normal vector \( \vec{n} = \langle A, B, C \rangle \) of the plane,
- \( D \) is a constant.

### Steps to Determine the Equation of a Plane:

1. **Identify the Normal Vector:**
   - You need a vector \( \vec{n} = \langle A, B, C \rangle \) that is perpendicular (normal) to the plane. If you have two vectors lying on the plane, you can find \( \vec{n} \) by taking their cross product.

2. **Find a Point on the Plane:**
   - Let's say you have a point \( P_0(x_0, y_0, z_0) \) that lies on the plane.

3. **Use the Point-Normal Form:**
   - The point-normal form of the plane equation is:
     \[
     A(x - x_0) + B(y - y_0) + C(z - z_0) = 0
     \]
   - Expanding this gives:
     \[
     Ax + By + Cz = D
     \]
   where \( D = Ax_0 + By_0 + Cz_0 \).

### Example:
Suppose you have:
- A point \( P(1, 2, 3) \) on the plane.
- A normal vector \( \vec{n} = \langle 2, -1, 4 \rangle \).

The equation becomes:
\[
2(x - 1) - 1(y - 2) + 4(z - 3) = 0
\]

Expanding it:
\[
2x - 2 - y + 2 + 4z - 12 = 0
\]
Simplifying further:
\[
2x - y + 4z = 12
\]

That's the equation of the plane.

Would you like further details or have any questions about this process? Here are some related topics that might interest you:

1. How to find the normal vector from three points on a plane.
2. The relation between the plane equation and the distance formula.
3. How to determine if a point lies on a given plane.
4. Deriving the plane equation using the vector form.
5. How to find the angle between two planes.

**Tip:** If you have three points on a plane, you can always find two vectors between them and use their cross product to determine the normal vector.

Learn how to write the equation of a plane in 3-dimensional space using the normal vector and a point on the plane. This guide explains key concepts like the normal vector, the point-normal form, and how to simplify the equation for any point on the plane. Suitable for Grades 10-12.

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