Question

Find the value of $\lambda$ are the vectors$\lambda \hat{i}+2\hat{j}+\hat{k}$, $3\hat{i}+2\hat{j}+\lambda \hat{k},\hat{i}+\hat{j}$ are coplanar.

Answer 1

Let the given vectors

$\begin{array}{c}\overrightarrow{a}=\lambda \hat{i}+2\hat{j}+\hat{k}\\ \\ \overrightarrow{b}=3\hat{i}+2\hat{j}+\lambda \hat{k}\\ \\ \overrightarrow{c}=\hat{i}+\hat{j}\end{array}$

Since they are coplanar

therefore,

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

$\begin{array}{c}\overrightarrow{b}\times \overrightarrow{c}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& 2& \lambda \\ 1& 1& 0\end{array}\right|\\ \\ =\hat{i}+\lambda \hat{j}+\hat{k}\end{array}$

Now,

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

$\begin{array}{c}\left(\lambda \hat{i}+2\hat{j}+\hat{k}\right).\left(\hat{i}+\lambda \hat{j}+\hat{k}\right)=0\\ \\ \lambda +2\lambda +1=0\\ \\ 3\lambda +1=0\\ \\ 3\lambda =-1\\ \\ \lambda =-\frac{1}{3}\end{array}$