Find an equation of the plane. The plane through the points (0, 5, 5), (5, 0, 5), and (5, 5, 0).

Solution:

Let (x₁, y₁, z₁) = (0, 5, 5)

(x₂, y₂, z₂) = (5, 0, 5)

(x₃, y₃, z₃) = (5, 5, 0)

We have equation of plane passing through three points as ,

\(\begin{vmatrix} x-x_{1} & y-y_{1} & z-z_{1} \\ x_{2} -x_{1} & y_{2}-y_{1} & z_{2}-z_{1}\\ x_{3} - x_{1}& y_{3}-y_{1} & z_{3}-z_{1} \end{vmatrix} = 0\)

\(\begin{vmatrix} x-0 & y-5 & z-5\\ 5-0 & 0-5 & 5-5\\ 5 - 0& 5-5 & 0-5 \end{vmatrix}=0\)

\(\begin{vmatrix} x-0 & y-5 & z-5 \\ 5 & -5 & 0\\ 5 & 0 & -5 \end{vmatrix}=0\)

⇒ x(25 - 0) - (y - 5)(-25 - 0) + (z -5)(0 + 25) = 0

⇒ 25x - (y - 5)(-25) + (z - 5)(25) = 0

Dividing throughout by 25 we get,

⇒ x + (y - 5) + (z - 5) = 0

⇒ x + y + z - 10 = 0

Find an equation of the plane. The plane through the points (0, 5, 5), (5, 0, 5), and (5, 5, 0).

Summary:

The equation of the plane through the points (0, 5, 5), (5, 0, 5), and (5, 5, 0) is x + y + z - 10 = 0.