Final answer:
To find the equation of a plane with a given point and a normal vector, use the point-normal form, substituting the coordinates of the point and the components of the normal vector into the equation: A(x - x_0) + B(y - y_0) + C(z - z_0) = 0.
Step-by-step explanation:
To find an equation of the plane that passes through a given point with a normal vector n, you can use the point-normal form of a plane equation. The point-normal form is:
A(x - x_0) + B(y - y_0) + C(z - z_0) = 0,
where (A, B, C) are the components of the normal vector n, and (x_0, y_0, z_0) are the coordinates of the given point through which the plane passes.
If the normal vector is not already normalized, it's important to use the actual vector given without normalizing it, as its length affects the equation of the plane. Each term in the equation represents the dot product of the normal vector with the vector connecting the given point to an arbitrary point (x, y, z) on the plane.
For example, if the normal vector is n = (2, -3, 5) and the plane passes through the point (1, -1, 2), then the equation of the plane is:
2(x - 1) - 3(y + 1) + 5(z - 2) = 0.