Find the general form of the equation of the plane passing through the three points. [Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.]
$$( - 1 , - 4 , - 2 ) , ( - 6,5,4 ) , ( 1 , - 2 , - 6 )$$
A) $12 x + 2 y + 7 z + 34 = 0$
B) $12 x + 2 y + 7 z = 0$
C) $x + 4 y + 2 z = 0$
D) $12 x + 2 y + 7 z - 34 = 0$
E) $x + 4 y + 2 z - 136 = 0$

Question

Quizplus · Accepted Answer

To find the equation of the plane passing through the points $(-1, -4, -2)$, $(-6, 5, 4)$, and $(1, -2, -6)$, we first find two vectors that lie in the plane by subtracting coordinates of these points. Let's use the first and second point to get one vector, and the first and third point to get another vector:Vector 1: $(-6 - (-1), 5 - (-4), 4 - (-2)) = (-5, 9, 6)$Vector 2: $(1 - (-1), -2 - (-4), -6 - (-2)) = (2, 2, -4)$Next, we find the normal vector to the plane by taking the cross product of these two vectors:$\mathbf{n} = \mathbf{v}_1 \times \mathbf{v}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ -5 & 9 & 6 \ 2 & 2 & -4 \end{vmatrix} = (9*(-4) - 6*2)\mathbf{i} - (-5*(-4) - 6*2)\mathbf{j} + (-5*2 - 9*2)\mathbf{k} = (-36 - 12)\mathbf{i} - (20 - 12)\mathbf{j} + (-10 - 18)\mathbf{k} = -48\mathbf{i} - 8\mathbf{j} - 28\mathbf{k}$Simplifying the coefficients by dividing by the greatest common divisor (4), we get the normal vector $\mathbf{n} = -12\mathbf{i} - 2\mathbf{j} - 7\mathbf{k}$.The general form of the equation of a plane is $Ax + By + Cz + D = 0$, where $(A, B, C)$ is the normal vector to the plane. So, we have $12x + 2y + 7z + D = 0$. To find $D$, we substitute one of the points into the equation. Using $(-1, -4, -2)$:$12(-1) + 2(-4) + 7(-2) + D = 0$$-12 - 8 - 14 + D = 0$$-34 + D = 0$$D = 34$Thus, the equation of the plane is $12x + 2y + 7z + 34 = 0$.

Find the General Form of the Equation of the Plane (−1,−4,−2),(−6,5,4),(1,−2,−6)( - 1 , - 4 , - 2 ) , ( - 6,5,4 ) , ( 1 , - 2 , - 6 )(−1,−4,−2),(−6,5,4),(1,−2,−6)

Find the General Form of the Equation of the Plane $( - 1 , - 4 , - 2 ) , ( - 6,5,4 ) , ( 1 , - 2 , - 6 )$