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!