• question_answer The intersection of the planes $2x-y-3z=8$ and $x+2y-4z=14$ is the line L. The value of 'a' for which the line L is perpendicular to the line through (a, 2, 2) and (6, 11, -1) is A)  10                                B)  9 C)  8                                 D)  5

Let $\vec{V}$ is the vector along the line of intersection of the planes $2x-y-3z-8=0$ and $x+2y-4z-14=0$ then $V=\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 2 & -1 & -3 \\ 1 & 2 & -4 \\ \end{matrix} \right|\,=5(\hat{i}+\hat{j}+\hat{k})$ This is $\bot$ to join of (a, 2, 2) and (6, 11, -1). Vector ${{\vec{V}}_{1}}$ along this is, ${{\vec{V}}_{1}}=(a-6)\hat{i}-9\hat{j}+3\hat{k}$ Now, $\vec{V}.\,{{\vec{V}}_{1}}=0$ gives a = 9.