Answer:
To find the set of all points equidistant from the points \(A(-1, 6, 2)\) and \(B(4, 3, -2)\), we can first find the midpoint of the line segment connecting points A and B, and then use that midpoint as the center of a sphere with a radius equal to half the distance between points A and B.
1. Find the midpoint M of line segment AB:
\[ M = \left(\frac{A_x + B_x}{2}, \frac{A_y + B_y}{2}, \frac{A_z + B_z}{2}\right) \]
\[ M = \left(\frac{-1 + 4}{2}, \frac{6 + 3}{2}, \frac{2 - 2}{2}\right) \]
\[ M = \left(\frac{3}{2}, \frac{9}{2}, 0\right) \]
2. Find the distance between points A and B, which will be the diameter of the sphere:
\[ AB = \sqrt{(B_x - A_x)^2 + (B_y - A_y)^2 + (B_z - A_z)^2} \]
\[ AB = \sqrt{(4 - (-1))^2 + (3 - 6)^2 + (-2 - 2)^2} \]
\[ AB = \sqrt{5^2 + (-3)^2 + (-4)^2} \]
\[ AB = \sqrt{25 + 9 + 16} \]
\[ AB = \sqrt{50} \]
\[ AB = 5\sqrt{2} \]
Now, we can write the equation of the sphere with center M and radius \(5\sqrt{2}/2\):
\[
(x - \frac{3}{2})^2 + (y - \frac{9}{2})^2 + (z - 0)^2 = (\frac{5\sqrt{2}}{2})^2
\]
Simplify the equation:
\[
(x - \frac{3}{2})^2 + (y - \frac{9}{2})^2 + z^2 = \frac{25}{2}
\]
So, the equation of the set of all points equidistant from points A and B is:
\[
(x - \frac{3}{2})^2 + (y - \frac{9}{2})^2 + z^2 = \frac{25}{2}
\]
This is the equation of a sphere with a diameter of AB, and its center is the midpoint of AB. The sphere is centered at \((\frac{3}{2}, \frac{9}{2}, 0)\) and has a radius of \(\frac{5\sqrt{2}}{2}\). The set of points described by this equation is a sphere in 3D space.
So, to answer your question, the set is a sphere with a diameter AB. It is not a cube or a plane.
Explanation: