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) \[{{\cos }^{-1}}\left( \frac{34}{63} \right)\]
Correct Answer: D
Solution :
[d] \[{{L}_{1}}={{\vec{r}}_{1}}=2\vec{i}+\vec{j}+2\vec{k}+t(-3\vec{i}+2\vec{j}+6\vec{k})\] \[{{L}_{2}}={{\vec{r}}_{2}}=(\vec{i}+2\vec{j}-\vec{k})+s(4\vec{i}-\vec{j}+8\vec{k})\] \[\therefore \] Angle between \[{{L}_{1}}\] and \[{{L}_{2}}\] is given by \[\cos \theta =\frac{(-3\vec{i}+2\vec{j}+6\vec{k}).(4\vec{i}-\vec{j}+8\vec{k})}{\sqrt{9+4+36}\sqrt{16+1+64}}\] \[=\frac{-12-2+48}{\sqrt{49}\sqrt{81}}=\frac{34}{7\times 9}=\frac{34}{63}\] \[\Rightarrow \theta ={{\cos }^{-1}}\left( \frac{34}{63} \right)\]
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