Question

Find $\lambda$ if the vectors $\overrightarrow{a}=\hat{i}+3\hat{j}+\hat{k}\overrightarrow{b}=2\hat{i}-\hat{j}-\hat{k}\overrightarrow{c}=\lambda \hat{i}+7\hat{j}+3\hat{k}$ are coplanar.

Answer 1

$a=i+3j+k$, $b=2i-j-k$, $c=\lambda i+7j+3k$

By determinant concept

$\left|\begin{array}{ccc}1& 3& 1\\ 2& -1& -1\\ \lambda & 7& 3\end{array}\right|=0$

$1(-3+7)-3(6+\lambda )+1(14+\lambda )=0$

$1\left(4\right)-18-3\lambda +14+\lambda =0$

$18-18-2\lambda =0$

$-2\lambda =0$

$\lambda =0$