The angle between the pair of lines given by \(\vec{r} = \hat{\imath}+2\hat{\jmath}-3\hat{k}+\lambda (\hat{\imath}-2\hat{\jmath}+2\hat{k})\) and \(\vec{r} = 5\hat{\imath}+\hat{\jmath}+\hat{k}+\mu (3\hat{\imath}-2\hat{\jmath} +6\hat{k})\)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\cos^{-1}\left(\displaystyle\frac{19}{21}\right)\)
The angle between the lines is the angle between their direction vectors, the vectors multiplied by \(\lambda\) and \(\mu\):
\(\vec{b}_1 = \hat{\imath} - 2\hat{\jmath} + 2\hat{k},\quad \vec{b}_2 = 3\hat{\imath} - 2\hat{\jmath} + 6\hat{k}\)
\(\vec{b}_1\cdot\vec{b}_2 = 3 + 4 + 12 = 19\)
\(|\vec{b}_1| = \sqrt{1 + 4 + 4} = 3,\quad |\vec{b}_2| = \sqrt{9 + 4 + 36} = 7\)
\(\displaystyle \cos\theta = \frac{\vec{b}_1\cdot\vec{b}_2}{|\vec{b}_1||\vec{b}_2|} = \frac{19}{21}\)
So \(\theta = \cos^{-1}\left(\frac{19}{21}\right)\). Options (1) and (4) are impossible, since \(\frac{21}{19} \gt 1\) and sine and cosine never exceed 1.
NCERT (2026-27 edition):
Angle between Two Lines
