Question

Find the value of λ so that the following vectors are coplanar:
(i) $\overrightarrow{a}=\hat{i}-\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+\hat{j}-\hat{k},\overrightarrow{c}=\lambda \hat{i}-\hat{j}+\lambda \hat{k}$

(ii) $\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}+2\hat{j}-3\hat{k},\overrightarrow{c}=\lambda \hat{i}+\lambda \hat{j}+5\hat{k}$

(iii) $\overrightarrow{a}=\hat{i}+2\hat{j}-3\hat{k},\overrightarrow{b}=3\hat{i}+\lambda \hat{j}+\hat{k},\overrightarrow{c}=\hat{i}+2\hat{j}+2\hat{k}$

(iv) $\overrightarrow{a}=\hat{i}+3\hat{j},\overrightarrow{b}=5\hat{k},\overrightarrow{c}=\lambda \hat{i}-\hat{j}$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{Given}: \\ \overrightarrow{a}=\hat{i}-\hat{j}+\hat{k} \\ \overrightarrow{b}=2\hat{i}+\hat{j}-\hat{k} \\ \overrightarrow{c}=\mathrm{\lambda }\hat{i}-j+\mathrm{\lambda }\hat{k} \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{vectors}\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\mathrm{are}\mathrm{coplanar}\mathrm{iff}\left[\overrightarrow{a}\mathit{}\overrightarrow{b}\mathit{}\overrightarrow{c}\right]=0. \\ \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\mathit{}\overrightarrow{a}\mathit{,}\mathit{}\overrightarrow{b}\mathit{,}\mathit{}\overrightarrow{c}\mathit{}\mathrm{are}\mathrm{coplanar}. \\ \therefore \left[\overrightarrow{a}\mathit{}\overrightarrow{b}\mathit{}\overrightarrow{c}\right]=0 \\ \Rightarrow \left|\begin{array}{ccc}1& -1& 1\\ 2& 1& -1\\ \mathrm{\lambda }& -1& \mathrm{\lambda }\end{array}\right|=0 \\ \Rightarrow 1\left(\mathrm{\lambda }-1\right)+1\left(2\mathrm{\lambda }+\mathrm{\lambda }\right)+1\left(-2-\mathrm{\lambda }\right)=0 \\ \Rightarrow \mathrm{\lambda }-1+3\mathrm{\lambda }-2-\mathrm{\lambda }=0 \\ \Rightarrow 3\mathrm{\lambda }-3=0 \\ \Rightarrow \mathrm{\lambda }=1 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{Given}: \\ \overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k} \\ \overrightarrow{b}=\hat{i}+2\hat{j}-3\hat{k} \\ \overrightarrow{c}=\mathrm{\lambda }\hat{i}+\mathrm{\lambda }j+5\hat{k} \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{vectors}\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\mathrm{are}\mathrm{coplanar}\mathrm{iff}\left[\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}\right]=0. \\ \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\mathit{}\overrightarrow{a}\mathit{,}\mathit{}\overrightarrow{b}\mathit{,}\mathit{}\overrightarrow{c}\mathrm{are}\mathrm{coplanar}. \\ \therefore \left[\overrightarrow{a}\mathit{}\overrightarrow{b}\mathit{}\overrightarrow{c}\right]=0 \\ \Rightarrow \left|\begin{array}{ccc}2& -1& 1\\ 1& 2& -3\\ \mathrm{\lambda }& \lambda & 5\end{array}\right|=0 \\ \Rightarrow 2\left(10+3\lambda \right)+1\left(5+3\lambda \right)+1\left(\lambda -2\lambda \right)=0 \\ \Rightarrow 8\lambda +25=0 \\ \Rightarrow \lambda =-\frac{25}{8} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\mathrm{Given}: \\ \overrightarrow{a}=\hat{i}+2\hat{j}-3\hat{k} \\ \overrightarrow{b}=3\hat{i}+\mathrm{\lambda }\hat{j}+\hat{k} \\ \overrightarrow{c}=\hat{i}+2\hat{j}+2\hat{k} \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{vectors}\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\mathrm{are}\mathrm{coplanar}\mathrm{iff}\left[\overrightarrow{a}\mathit{}\overrightarrow{b}\mathit{}\overrightarrow{c}\right]=0. \\ \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\mathit{}\overrightarrow{a}\mathit{,}\mathit{}\overrightarrow{b}\mathit{,}\mathit{}\overrightarrow{c}\mathit{}\mathrm{are}\mathrm{coplanar}. \\ \therefore \left[\overrightarrow{a}\mathit{}\overrightarrow{b}\mathit{}\overrightarrow{c}\right]=0 \\ \Rightarrow \left|\begin{array}{ccc}1& 2& -3\\ 3& \lambda & 1\\ 1& 2& 2\end{array}\right|=0 \\ \Rightarrow 1\left(2\lambda -2\right)-2\left(6-1\right)-3\left(6-\lambda \right)=0 \\ \Rightarrow 5\lambda -30=0 \\ \Rightarrow \mathrm{\lambda }=6 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right)\mathrm{Given}: \\ \overrightarrow{a}=\hat{i}+3\hat{j} \\ \overrightarrow{b}=5\hat{k} \\ \overrightarrow{c}=\mathrm{\lambda }\hat{i}-\hat{j} \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{vectors}\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\mathrm{are}\mathrm{coplanar}\mathrm{iff}\left[\overrightarrow{a}\mathit{}\overrightarrow{b}\mathit{}\overrightarrow{c}\right]=0. \\ \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\mathit{}\overrightarrow{a}\mathit{,}\mathit{}\overrightarrow{b}\mathit{,}\mathit{}\overrightarrow{c}\mathit{}\mathrm{are}\mathrm{coplanar}. \\ \therefore \left[\overrightarrow{a}\mathit{}\overrightarrow{b}\mathit{}\overrightarrow{c}\right]=0 \\ \Rightarrow \left|\begin{array}{ccc}1& 3& 0\\ 0& 0& 5\\ \mathrm{\lambda }& -1& 0\end{array}\right|=0 \\ \Rightarrow 1\left(0+5\right)-3\left(0-5\lambda \right)+0\left(0-0\right)=0 \\ \Rightarrow 5+15\lambda =0 \\ \Rightarrow \mathrm{\lambda }=-\frac{1}{3} \end{gathered}$$