The angle between the lines \(\vec{r} = (\hat{\imath}+\hat{\jmath})+\lambda(2\hat{\imath}-2\hat{\jmath}+\hat{k})\) and \(\vec{r} = (3\hat{\imath}-\hat{k})+\mu(6\hat{\imath}+3\hat{\jmath}-2\hat{k})\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\cos^{-1}\left(\displaystyle\frac{4}{21}\right)\)
The angle between the lines is the angle between their directions \(\vec{b}_1 = 2\hat{\imath}-2\hat{\jmath}+\hat{k}\) and \(\vec{b}_2 = 6\hat{\imath}+3\hat{\jmath}-2\hat{k}\):
\(\vec{b}_1\cdot\vec{b}_2 = 12-6-2 = 4\)
\(|\vec{b}_1| = \sqrt{4+4+1} = 3\)
\(|\vec{b}_2| = \sqrt{36+9+4} = 7\)
\( \displaystyle \cos\theta = \frac{|\vec{b}_1\cdot\vec{b}_2|}{|\vec{b}_1|\,|\vec{b}_2|} = \frac{4}{21} \)
So \(\theta = \cos^{-1}\left(\frac{4}{21}\right)\). The points \(\hat{\imath}+\hat{\jmath}\) and \(3\hat{\imath}-\hat{k}\) play no part.