Question

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

$(6,-4,4),(0,0,-4),(-1,-2,-3),(1,2,-5)$

Answer 1

Let

$A=(6,-4,4)$
$B=(0,0,-4)$
$C=(-1,-2,-3)$
$D=(1,2,-5)$

Then
$AB=\overline{a}=-6i+4j-8k$
$BC=\overline{b}=-i-2j+k$
$CD=\overline{c}=2i+4j-2k$

Now

$(\overline{a}\times \overline{b})=(-6i+4j-8k)\times (-i-2j+k)=20i+14j-8k$

Now

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

$=(2i+4j-2k).(-12i+14j+16k)$
$=-24+56-32$
$=-56+56$
$=0$

Hence they are co-planar.

Hence the answer is 1.