Question

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

$(3,2,-5),(-3,8,-5)(-3,2,1),(-1,4,-3).$

Answer 1

Let,

$A=(3,2,-5)$

$B=(-3,8,-5)$

$C=(-3,2,1)$

$D=(-1,4,-3)$

Then,

$AB=\overline{a}=-6i+6j-0k$

$BC=\overline{b}=0i-6j+6k$

$CD=\overline{c}=2i+2j-4k$

Now

$(\overline{a}\times \overline{b})=(-6i+6j-0k)\times (0i-6j+6k)=36i+36j+36k$

Now

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

$=(2i+2j-4k).(36i+36j+36k)$

$=72+72-144$

$=-144+144$

$=0$

Hence they are co-planar.

Hence the answer is 1.