Find the equation of the plane that passes th | vedclass

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Question

(A) Let the given points be $A(1,1,0), B(1,2,1),$ and $C(-2,2,-1)$.The equation of a plane passing through three points $(x_1, y_1, z_1), (x_2, y_2, z

Vedclass · Accepted Answer

$2x + 3y - 3z = 5$

Vedclass · Answer

(A) Let the given points be $A(1,1,0), B(1,2,1),$ and $C(-2,2,-1)$.The equation of a plane passing through three points $(x_1, y_1, z_1), (x_2, y_2, z_2),$ and $(x_3, y_3, z_3)$ is given by the determinant equation:$\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \ x_2-x_1 & y_2-y_1 & z_2-z_1 \ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0$Substituting the coordinates of points $A, B,$ and $C$:$\begin{vmatrix} x-1 & y-1 & z-0 \ 1-1 & 2-1 & 1-0 \ -2-1 & 2-1 & -1-0 \end{vmatrix} = 0$$\begin{vmatrix} x-1 & y-1 & z \ 0 & 1 & 1 \ -3 & 1 & -1 \end{vmatrix} = 0$Expanding along the first row:$(x-1)(-1-1) - (y-1)(0 - (-3)) + z(0 - (-3)) = 0$$(x-1)(-2) - (y-1)(3) + z(3) = 0$$-2x + 2 - 3y + 3 + 3z = 0$$-2x - 3y + 3z + 5 = 0$Multiplying by $-1$,we get:$2x + 3y - 3z = 5$

Find the equation of the plane that passes through the three points $(1,1,0), (1,2,1),$ and $(-2,2,-1)$.

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