Question

Examine if the following sets of points are coplanar: If they are coplanar write 1 otherwise write 0.

$(-2,-1,0),(1,-2,-1),(2,1,4),(0,1,0)$

Answer 1

Let

$A=(-2,-1,0)$
$B=(1,-2,-1)$
$C=(2,1,4)$
$D=(0,1,0)$

Then
$AB=\overline{a}=3i-j-k$
$BC=\overline{b}=i+3j+5k$
$CD=\overline{c}=-2i+0j-4k$

Now

$(\overline{a}\times \overline{b})=(3i-j-k)\times (i+3j+5k)=-2i-16j+10k$

Now

$\overline{c}.(\overline{a}\times \overline{b})$

$=(-2i+0j-4k).(-2i-16j+10k)$
$=4-40$
$=-36$
$\ne 0$

Hence they are non-co-planar.

Hence the answer is 0