Vector Algebra TG EAPCET Previous Year Chapters wise Questions

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The position vectors and direction vectors for two lines are given:Line $L_1: r=a+tb$, where $a=\hat{i}-\hat{j}+3\hat{k}$ and $b=2\hat{i}-\hat{j}+\lambda\hat{k}$.Line $L_2: r=c+sd$, where $c=-\hat{k}$ and $d=\hat{i}+2\hat{j}-\hat{k}$.To determine whether the lines are coplanar or skew, we set up equations based on their parametric forms:$L_1\;=\begin{pmatrix}1\\-1\\3\end{pmatrix}+t\begin{pmatrix}2\\-1\\lambda\end{pmatrix}$$L_2\;=\begin{pmatrix}0\\0\\-1\end{pmatrix}+s\begin{pmatrix}1\\2\\-1\end{pmatrix}$By equating the components of the parametric equations, we get:$\;1+2t=s$$\;-1-t=2s$$\;3+\lambda t=-1-s$Solving the first two equations simultaneously gives:$\;2+4t=2s$$\;-1-t=2s$$\;\frac{-1-t}{2s}=\;\frac{3+5t}{0}$From these equations, solve for $t$ and $s$ :$\Rightarrow t=-\frac{3}{5}$$\Rightarrow s=-\frac{1}{5}$ after substituting $t$.For the third equation:$3-\frac{3\lambda}{5}=-1+\frac{1}{5}$From this, solve for $\lambda$ :$\;21-\frac{1}{5}=\frac{3\lambda}{5}$$\;\Rightarrow \lambda=\frac{19}{3}$Thus, the lines are skew when $\lambda \neq \frac{19}{3}$. If $\lambda=\frac{19}{3}$, the lines would be coplanar.