Final answer:
To find the plane equation that passes through the given points, calculate vectors in the plane from the point pairs, compute their cross product to find the normal vector, and then use this vector with any of the points to establish the plane equation: -25x - 25y + 25z = -250.
Step-by-step explanation:
To find an equation of the plane that passes through the points (0, 5, 5), (5, 0, 5), and (5, 5, 0), we need to determine the normal vector to the plane and then use it in the plane equation format.
Step 1: Find Two Vectors in the Plane
We begin by finding two vectors that lie in the plane by subtracting the coordinates of two pairs of the given points:
- Vector A: (5, 0, 5) - (0, 5, 5) = (5, -5, 0)
- Vector B: (5, 5, 0) - (0, 5, 5) = (5, 0, -5)
Step 2: Calculate the Normal Vector
We calculate the cross product of vectors A and B to get the normal vector N:
N = A × B = (5, -5, 0) × (5, 0, -5) = (-25, -25, 25)
Step 3: Find the Plane Equation
The equation of the plane is given by:
N · (x, y, z) = N · (x0, y0, z0)
Substituting N and any point, such as (0, 5, 5), we get:
(-25, -25, 25) · (x - 0, y - 5, z - 5) = 0
This simplifies to the plane equation:
-25x - 25y + 25z = -250