Question

The coordinates of the points A, B, C, D are $(4,\alpha ,2)$, $(5,-3,2)$, $(\beta ,1,1)$ & $(3,3,-1)$. Line AB would be perpendicular to line CD when?

• A. $\alpha =-1,\beta =-1$
• B. $\alpha =1,\beta =2$
• C. $\alpha =2,\beta =1$
• D. $\alpha =2,\beta =2$

Answer 1

The correct option is A $\alpha =-1,\beta =-1$

$A(4,\alpha ,2),B(5,-3,2),C(\beta ,1,1),D(3,3,-1)\therefore \overrightarrow{a}=4\hat{i}+\alpha \hat{j}+2\hat{k},\overrightarrow{b}=5\hat{i}-3\hat{j}+2\hat{k},\overrightarrow{c}=\beta \hat{i}+\hat{j}+\hat{k},\overrightarrow{d}=3\hat{i}+3\hat{j}-\hat{k}\therefore \overrightarrow{AB}=\hat{i}+(-3-\alpha )\hat{j}\therefore \overrightarrow{CD}=(3-\beta )\hat{i}+2\hat{j}-2\hat{k}\therefore \overrightarrow{AB}.\overrightarrow{CD}=0(\because ABperpCD)[\hat{i}-(3+\alpha )\hat{j}].[(3-\beta )\hat{i}+2\hat{j}-2\hat{k}]=0(3-\beta )+2(-3-\alpha )=03-\beta -6-2\alpha =0\Rightarrow \beta +2\alpha =-3$

Option A satisfies