The angle between the pairs of lines
\(\vec{r} = (3\hat{\imath} + 2\hat{\jmath} - 4\hat{k}) + \lambda(\hat{\imath} + 2\hat{\jmath} + 2\hat{k})\)
\(\vec{r} = (5\hat{\imath} - 2\hat{\jmath}) + \mu(3\hat{\imath} + 2\hat{\jmath} + 6\hat{k})\) is :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\theta = \cos^{-1}\left(\displaystyle\frac{19}{21}\right)\)
The angle between the lines is the angle between their direction vectors:
\(\vec{b}_1 = \hat{\imath} + 2\hat{\jmath} + 2\hat{k}\)
\(\vec{b}_2 = 3\hat{\imath} + 2\hat{\jmath} + 6\hat{k}\)
\(\vec{b}_1 \cdot \vec{b}_2 = 3 + 4 + 12 = 19\)
\(|\vec{b}_1| = \sqrt{1 + 4 + 4} = 3\)
\(|\vec{b}_2| = \sqrt{9 + 4 + 36} = 7\)
So
\( \displaystyle \cos\theta = \frac{\vec{b}_1 \cdot \vec{b}_2}{|\vec{b}_1|\,|\vec{b}_2|} = \frac{19}{3 \times 7} = \frac{19}{21} \)
\( \displaystyle \theta = \cos^{-1}\left(\frac{19}{21}\right) \)
NCERT (2026-27 edition):
Angle between Two Lines
