Question

Determine whether the given points are coplanar. ​A(44​,44​,44​), ​B(11​,22​,11​), ​C(55​,11​,44​), ​D(22​,33​,77​)

Answer 1

Answer:

No

Step-by-step explanation:

Let given points are A (44,44,44),B(11,22,11) ,C(55,11,44) and D(22,33,77).

We have to determine the points are co-planar or not.

$AB=B-A=(11,22,11)-(44,44,44)=(-33,-22,-33)$

$BC=C-A=(55,11,44)-(11,22,11)=(44,-11,33)$

$CD=D-C=(22,33,77)-(55,11,44)=(-33,22,33)$

When three vectors A, B and C are co-planar then

$(A\times B)\cdot C=0$

Using the formula

$AB\times BC\times CD=\begin{vmatrix}-33&-22&-33\\44&-11&33\\-33&22&33\end{vmatrix}$

$AB\times BC\cdot CD=-33(-363-726)+22(1452+1089)-33(968-363)=71874$

$AB\times BC\cdot CD\neq 0$

Therefore, A, B,C and D are not co-planar.