Question

Find the value of $\lambda$ for which the four points with position vectors $6\hat{i}-7\hat{j},16\hat{i}-19\hat{j}-4\hat{k},\lambda \hat{j}-6\hat{k}$ and $2\hat{i}-5\hat{j}+10\hat{k}$ are coplanar.

Answer 1

Let $\overrightarrow{A}=6i-7j,\overrightarrow{B}=16-19j-4k=0,\overrightarrow{C}=\lambda j-6k$ and $\overrightarrow{D}=2i-5j+10k$

$\overrightarrow{AB}=10i-12j-4k\overrightarrow{AC}=-6i+(\lambda +7)j-6k\overrightarrow{AD}=-4i+2j+10k$

Now the vectors are coplanar

$\Rightarrow \overrightarrow{AB}.(\overrightarrow{AC}\times \overrightarrow{AD})=0\Rightarrow \left|\begin{array}{ccc}10& -12& -4\\ -6& \lambda +7& -6\\ -4& 2& 10\end{array}\right|=0\Rightarrow 10(10\lambda +70+12)+12(-60-24)-4(-12+4\lambda +28)=0\Rightarrow 100\lambda +820-1008-64-16\lambda =0\Rightarrow 84\lambda =252\Rightarrow \lambda =3$