The acute angle between the lines \(\vec{r} = (4\hat{\imath} -\hat{\jmath}) +\lambda(2\hat{\imath} +\hat{\jmath}-3\hat{k})\) and \(\displaystyle\frac{x - 1}{1} = \frac{y + 1}{-3} = \frac{z - 2}{2}\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{\pi}{3}\)
The direction vectors of the two lines are
\(\vec{b}_1 = 2\hat{\imath} + \hat{\jmath} - 3\hat{k},\quad \vec{b}_2 = \hat{\imath} - 3\hat{\jmath} + 2\hat{k}\)
\(\vec{b}_1\cdot\vec{b}_2 = 2 - 3 - 6 = -7\)
\(|\vec{b}_1| = |\vec{b}_2| = \sqrt{4 + 1 + 9} = \sqrt{14}\)
For the acute angle, take the absolute value of the dot product:
\(\displaystyle \cos\theta = \frac{|\vec{b}_1\cdot\vec{b}_2|}{|\vec{b}_1|\,|\vec{b}_2|} = \frac{7}{14} = \frac{1}{2}\)
\(\displaystyle \theta = \frac{\pi}{3}\)