Question

Find the value of $\lambda$ for which the vectors $2\hat{i}-4\hat{j}+5\hat{k}$, $\hat{i}-\lambda \hat{j}+\hat{k}$ and $3\hat{i}+2\hat{j}-5\hat{k}$ are coplanar.

Answer 1

If three vectors are coplanar then their scalar product is 0.

$\overrightarrow{a}.[\overrightarrow{b}\times \overrightarrow{c}]=0$

$\left|\begin{array}{ccc}2& -4& 5\\ 1& -\lambda & 1\\ 3& 2& -5\end{array}\right|=0$

$⟹2[5\lambda -2]+4[-5-3]+5[2+3\lambda ]=0$

$⟹10\lambda -4-32+10+15\lambda =0$

$⟹25\lambda -26=0$

$⟹\lambda =\frac{26}{25}$