Question

Find the value of $\lambda$ for which $\text{A (3, 2, 1), B(4, 2, - 2), C (6, 5, - 1)}$ and $\text{D (}\lambda \text{, 5, 5)}$ are
coplanar.

Answer 1

If $A,B,C,D$ are coplanar vectors, then

$[\overrightarrow{AB},\overrightarrow{AC},\overrightarrow{AD}]=0$

$\overrightarrow{AB}=(4-3)\hat{i}+(2-2)\hat{j}+(-2-1)\hat{k}=\hat{i}-3\hat{k}$

$\overrightarrow{AC}=(6-3)\hat{i}+(5-2)\hat{j}+(-1-1)\hat{k}=3\hat{i}+3\hat{j}-2\hat{k}$

$\overrightarrow{AD}=(\lambda -3)\hat{i}+(5-2)\hat{j}+(5-1)\hat{k}=(\lambda -3)\hat{i}+3\hat{j}+4\hat{k}$

Since they are coplanar,

$\left|\begin{array}{ccc}1& 0& -3\\ 3& 3& -2\\ \lambda -3& 3& 4\end{array}\right|$

$\Rightarrow 1(12+6)-0-3(9-3\lambda +9)=0$

$\Rightarrow 18-3(18-3\lambda )=0$

$\Rightarrow 18-9(6-\lambda )=0$

$\Rightarrow 2-6+\lambda =0$

$\therefore \lambda =4$