A) \[{{\cos }^{-1}}\left( \frac{\sqrt{41}}{34} \right)\]
B) \[{{\cos }^{-1}}\left( \frac{21}{34} \right)\]
C) \[{{\cos }^{-1}}\left( \frac{43}{63} \right)\]
D) \[{{\cos }^{-1}}\left( \frac{5\sqrt{23}}{41} \right)\]
E) \[{{\cos }^{-1}}\left( \frac{34}{63} \right)\]
Correct Answer: E
Solution :
Given lines can be rewritten as \[\overrightarrow{r}=2\hat{i}+\hat{j}+2\hat{k}+t(-3\hat{i}+2\hat{j}+6\hat{k})\] and \[\overrightarrow{r}=\hat{i}+2\hat{j}-\hat{k}+s(4\hat{i}-\hat{j}+8\hat{k})\] Here, \[{{a}_{1}}=-3,{{b}_{1}}=2,{{c}_{1}}=6\] and \[{{a}_{2}}=4,{{b}_{2}}=-1,{{c}_{2}}=8\] \[\therefore \]\[\cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\] \[=\frac{-3\times 4+2\times (-1)+6\times 8}{\sqrt{9+4+36}\sqrt{16+1+64}}\] \[=\frac{34}{7\times 9}\] \[\Rightarrow \] \[\theta ={{\cos }^{-1}}\left( \frac{34}{63} \right)\]
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