Find an equation of the set of all points equidistant from the points A(1, 5, 3) and B(6, 2, 2). Describe the set.

To find the equation of the set of points that are equidistant from the two points A(1, 5, 3) and B(6, 2, 2), we need to determine the set of points P(x, y, z) that satisfy the condition:

Distance from A to P = Distance from B to P

This can be expressed mathematically as follows:

sqrt((x - 1)² + (y - 5)² + (z - 3)²) = sqrt((x - 6)² + (y - 2)² + (z - 2)²)

To eliminate the square roots, we can square both sides of the equation:

(x - 1)² + (y - 5)² + (z - 3)² = (x - 6)² + (y - 2)² + (z - 2)²

Now expanding both sides:

(x² - 2x + 1 + y² - 10y + 25 + z² - 6z + 9) = (x² - 12x + 36 + y² - 4y + 4 + z² - 4z + 4)

After simplifying, we have:

-2x + 1 - 10y + 25 - 6z + 9 = -12x + 36 - 4y + 4 - 4z + 4

This leads to:

10x - 6y - 2z + 3 = 0

This equation represents a plane in three-dimensional space. To describe the set, it consists of all points (x, y, z) that lie on this plane, which are exactly halfway between points A and B in terms of their distance.