Find the value of if four points with position vectors

Find the value of $\lambda ,$ if four points with position vectors $3\hat{i}+6\hat{j}+9\hat{k},\hat{i}+2\hat{j}+3\hat{k},\; 2\hat{i}+3\hat{j}+\hat{k}$ and $4\hat{i}+6\hat{j}+\lambda \hat{k}$ are coplanar.

Answers (1)

Let the four points be A, B, C and D with position vectors $3\hat{i}+6\hat{j}+9\hat{k}, \hat{i}+2\hat{j}+3\hat{k}, 2\hat{i}+3\hat{j}+\hat{k}\, and\: 4\hat{i}+6\hat{j}+\lambda \hat{k}\; resp\cdot$
Then $\overrightarrow{OA}= 3\hat{i}+6\hat{j}+9\hat{k}$
         $\overrightarrow{OB}= \hat{i}+2\hat{j}+3\hat{k}$
          $\overrightarrow{OD}= 4\hat{i}+6\hat{j}+\lambda \hat{k}$
Then $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}= \left ( \hat{i} +2\hat{j}+3\hat{k}\right )-\left ( 3\hat{i}+6\hat{j}+9\hat{k} \right )$
                                             $= -2\hat{i}-4\hat{j}-6\hat{k}$
$\overrightarrow{AC}= \overrightarrow{OC}- \overrightarrow{OA}= \left ( 2\hat{i}+3\hat{j}+\hat{k} \right )-\left ( 3\hat{i}+6\hat{j} +9\hat{k}\right )$
                                     $= -\hat{i}-3\hat{j}-8\hat{k}$
and $\overrightarrow{AD}= \overrightarrow{OD}-\overrightarrow{OA}= \left ( 4\hat{i} +6\hat{j}+\lambda \hat{k}\right )-\left ( 3\hat{i}+6\hat{j}+9\hat{k} \right )$
                                             $= \hat{i}+\left ( \lambda -9 \right )\hat{k}$
Since the points are coplanar, then
$\left [ \overrightarrow{AB},\overrightarrow{AC},\overrightarrow{AD} \right ]= 0$
i.e $\begin{bmatrix} -2 & -4 & -6\\ -1&-3 &-8 \\ 1&0 & \lambda -9 \end{bmatrix}= 0$
$\Rightarrow -2\left ( -3\lambda +27 \right )+4\left ( -\lambda +9+8 \right )-6\left ( 3 \right )= 0$
$\Rightarrow 6\lambda -54-4\lambda +68-18= 0$
$\Rightarrow 2\lambda -4= 0$
$\Rightarrow \lambda= 2$