Question

Show that the four points $A(4,5,1),B(0,-1,-1),C(3,9,4)$ and $D(-4,4,4)$ are coplanar.

Answer 1

For the four points $A,B,C,D$ to be coplanar , they should lie on same plane which implies $AD.(AC\times AB)=0$
$AB=4i+6j+2k$ , $AC=i-4j-3k$ and $AD=8i+j-3k$
$AB\times AC=i(-18+8)+j(2+12)+k(-16-6)=-10i+14j-22k$
$AD.(AB\times AC)=(8i+j-3k).(-10i+14j-22k)=-80+14+66=0$
Therefore points $A,B,C,D$ are coplanar.