https://file.aiquickdraw.com/mathbot/file/1725909392625h6rwylji.jpg

so for any x, y, z, the plane will contain the points x_0, y_0, and z_0? 

The equation shown in the image is:

\[
A(x - x_0) + B(y - y_0) + C(z - z_0) = 0
\]

This is the equation of a plane in three-dimensional space. The constants \( A \), \( B \), and \( C \) define the direction of the normal vector to the plane. The point \( (x_0, y_0, z_0) \) is a specific point on the plane.

### Key Insight:

Yes, any point \( (x_0, y_0, z_0) \) will lie on the plane. This is because if you substitute \( x = x_0 \), \( y = y_0 \), and \( z = z_0 \) into the equation, the equation simplifies to \( 0 = 0 \), which is true.

### To clarify:

- For any other point \( (x, y, z) \), the left-hand side will evaluate to some value based on the normal vector, and if it equals zero, that point lies on the plane.
- If the equation does not equal zero for a point \( (x, y, z) \), then that point does not lie on the plane.

Would you like more details on how planes work or their geometric interpretation?

### Related Questions:
1. What is the geometric meaning of the normal vector \( (A, B, C) \)?
2. How can we derive the distance of a point from a plane using this equation?
3. Can you show how to find the equation of a plane given three points?
4. What happens if \( A = B = C = 0 \) in the plane equation?
5. How can we find the intersection between two planes?

#### Tip:
To find out whether a point lies on a plane, substitute the coordinates of the point into the plane's equation and see if it satisfies the equation!

so for any x, y, z, the plane will contain the points x_0, y_0, and z_0?

Explore how the equation of a plane in three-dimensional space ensures that any point (x_0, y_0, z_0) lies on the plane. This topic covers essential geometric and vector concepts suitable for high school students.

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