Find an equation of the plane. The plane through the points are (0, 7, 7), (7, 0, 7), and (7, 7, 0).

Solution:

Let (x\(_1\), y\(_1\), z\(_1\)) = (0, 7, 7)

(x\(_2\), y\(_2\), z\(_2\))= (7, 0, 7)

(x\(_3\), y\(_3\), z\(_3\)​​​​​​​)= (7, 7, 0)

We have equation of plane passing through three points as,

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

\(\begin{vmatrix} x-0 & y-7 & z-7 \\ 7-0 & 0-7 & 7-7\\ 7 - 0& 7-7 & 0-7 \end{vmatrix}=0\)

\(\begin{vmatrix} x-0 & y-7 & z-7 \\ 7 & -7 & 0\\ 7 & 0 & -7 \end{vmatrix}=0\)

⇒ x(49 - 0) - (y - 7)(-49 - 0) + (z - 7)(0 + 49) = 0

⇒ 49x - (y - 7)(-49) + (z - 7)(49) = 0

Dividing throughout by 49 we get,

⇒ x + (y - 7) + (z - 7) = 0

⇒ x + y + z - 14 = 0

Find an equation of the plane. The plane through the points are (0, 7, 7), (7, 0, 7), and (7, 7, 0).

Summary:

The equation of the plane through the points (0, 7, 7), (7, 0, 7), and (7, 7, 0) is x + y + z - 14 = 0.