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.