Answer to Question #103625 in Analytic Geometry for ag

Question

Check whether the points (1,−1,−2),(1,−4,2),(3,0,2),(4,−3−2) are coplanar
or not. If they are coplanar, write the equation of the plane they pass through.
Otherwise, change the coordinates of one of the points so that they become
coplanar. In this case, find the plane passing through them.

Answer 1

let's name the points

A(1,−1,−2); B(1,−4,2); C(3,0,2); D(4,−3−2)

then compute vectors BA, CA, DA:

BA={0,3,-4};CA={-2,-1,-4};DA={-3,2,0}

and compute  the triple product of these vectors:

$\begin{vmatrix} 0 & 3 & -4 \\ -2 & -1 & -4 \\ -3 & 2 & 0 \end{vmatrix}$ ​ =0*(-1*0-(-4*2))-3*(-2*0-(-4*(-3)))-4*(-2*2-(-3)*(-1))=64

then, if the triple product is not equal to zero, then the vectors are not coplanar, then the points are not coplanar

we will change coordinates B to (6,-2,2) , then check if they coplanar again

BA={-5,1,-4};CA={-2,-1,-4};DA={-3,2,0}

$\begin{vmatrix} -5 & 1 & -4 \\ -2 & -1 & -4 \\ -3 & 2 & 0 \end{vmatrix}$ =0, so they are coplanar now,

The plane, coplanar to 2 vectors (CA, DA) and passing through point A will be

$\begin{vmatrix} x-1 & y+1 & z+2 \\ -2 & -1 & -4 \\ -3 & 2 & 0 \end{vmatrix}$ =0 OR 8x+12y-7z-10=0