Question

Find an equation for the plane through the three points p(0, 1, 1), q(1, 0, 1), and r(1, 1, 0)

Answer 1

The equation for the plane through the points p(0, 1, 1), q(1, 0, 1) and r(1, 1, 0) is x + y + z = 2

Given,

Three points :- p(0, 1, 1), q(1, 0, 1), r(1, 1, 0)

We can take this as ($(x_{1} ,y _{1}, z_{1} ), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3}, z_{3})$ respectively.

Thus,

The equation for the plane through the points

Here,

The formula of equation of the plane passing through three non collinear points

$x-x_{1} y-y_{1} z-z_{1} \\x_{2} -x_{1} y_{2} -y_{1} z_{2} -z_{1} = 0\\x_{3} -x_{2} y_{3} -y_{2} z_{3} -z_{2}$

by substituting the values in the formula of equation of the plane

x-0  y-1  z-1

1-0  0-1  1-1    = 0

1-0  1-1  0-1

x  y  z

1  -1  0   =  0

1  0  -1

x(1) - (y-1)(-1) - (z-1)(-1) = 0

x + y - 1 + z - 1 = 0

x + y + z + 2 = 0

x + y + z = 2

The equation of the plane through the points p(0, 1, 1), q(1, 0, 1), and r(1, 1, 0) is x + y + z = 2

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Answer 2

The equation of the plane through the three points p(0, 1, 1), q(1, 0, 1), and r(1, 1, 0) is x + y + z = 2

For given question,

We have been given three points p(0, 1, 1), q(1, 0, 1), and r(1, 1, 0)

We need to find an equation for the plane through the given three points.

We know that the equation of the plane through the points $(x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3)$ is,

$\begin{vmatrix}x-x_1 & y-y_1 & z-z_1\\x_2-x_1 & y_2-y_1 & z_2-z_1\\x_3-x_1 & y_3-y_1 & z_3-z_1\end{vmatrix}=0$

Assume that for given points,

$(x_1,y_1,z_1)=(0, 1, 1)\\\\(x_1,y_1,z_1)=(1, 0, 1)\\\\(x_1,y_1,z_1)=(1, 1, 0)$

So, the equation of the plane passing though given points would be,

$\Rightarrow \begin{vmatrix}x-0 & y-1 & z-1\\1-0 & 0-1 & 1-1\\1-0 & 1-1 & 0-1\end{vmatrix}=0\\\\\\\Rightarrow \begin{vmatrix}x-0 & y-1 & z-1\\1 & -1 & 0\\1 & 0 & -1\end{vmatrix}=0\\\\\\\Rightarrow x(1-0)-(y-1)(-1-0)+(z-1)(1-0)=0\\\\\Rightarrow x+y-1+z-1=0\\\\\Rightarrow x+y+z=2$

Therefore, the equation of the plane through the three points p(0, 1, 1), q(1, 0, 1), and r(1, 1, 0) is x + y + z = 2

Learn more about the equation of the plane here:

brainly.com/question/27190150

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