Question

Show that the points $A(3,9,4),B(0,-1,-1)C(-4.4,4)$ and $D(4,5,1)$ are coplaner.

Answer 1

Let $\overrightarrow{a}=\overrightarrow{AB},\overrightarrow{b},=\overrightarrow{AC},\overrightarrow{c}=\overrightarrow{AD}$

$\overrightarrow{a}=-3\overrightarrow{i}-10\overrightarrow{j}-5\overrightarrow{k}$

$\overrightarrow{b}=-7\overrightarrow{i}-5\overrightarrow{j}-10\overrightarrow{k}$

$\overrightarrow{c}=\overrightarrow{i}-4\overrightarrow{j}-3\overrightarrow{k}$

We know that these vector are coplaner, it $\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c}=0)$

$\overrightarrow{b}\times \overrightarrow{c}=$

$\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ -7& -5& 0\\ 1& -4& -3\end{array}\right|$

$=15\overrightarrow{i}-21\overrightarrow{j}+33\overrightarrow{k}$

$=3(5\overrightarrow{i}-7\overrightarrow{j}+11\overrightarrow{k})$

$\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=0$

Hence coplaner