If the line with direction ratios $(1, \alpha, \beta)$ is perpendicular to the line with direction ratios…
If the line with direction ratios $(1, \alpha, \beta)$ is perpendicular to the line with direction ratios $(-1,2,1)$ and parallel to the line with direction ratios $(\alpha, 1, \beta)$, then $(\alpha, \beta)$ is
$(-1,-1)$
$(1,-1)$
$(-1,3)$
$(1,1)$
Solution
Since line with $d r^{\prime} s(1, u, \beta)$ is perpendicular to line with $d r^{\prime} s(-1,2,1)-1+2 \alpha+\beta=0 \Rightarrow 2 \alpha+\beta=1$
Also, $\lambda(\hat{i}+\alpha \tilde{j}+\beta \hat{k})=\alpha \hat{i}+\hat{j}+\beta \vec{k}$ $\lambda=\alpha$
and $\lambda \beta=\beta \Rightarrow \lambda=1 \Rightarrow \alpha=1, \beta=-1$ $(\alpha, \beta)=(1,-1)$.