The angle between the lines \(\vec{r} = 3\hat{\imath} - 2\hat{\jmath} + 1\hat{k} + \mu\left(4\hat{\imath} + 6\hat{\jmath} + 12\hat{k}\right)\) and \(\vec{r} = 7\hat{\imath} - 3\hat{\jmath} + 9\hat{k} + \lambda\left(5\hat{\imath} + 8\hat{\jmath} - 4\hat{k}\right)\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\cos^{-1}\displaystyle\frac{10}{7\sqrt{105}}\)
The angle between two lines is the angle between their direction vectors:
\(\vec{b}_1 = 4\hat{\imath} + 6\hat{\jmath} + 12\hat{k}\)
\(\vec{b}_2 = 5\hat{\imath} + 8\hat{\jmath} - 4\hat{k}\)
\(\vec{b}_1\cdot\vec{b}_2 = 20 + 48 - 48 = 20\)
\(|\vec{b}_1| = \sqrt{16 + 36 + 144} = \sqrt{196} = 14\)
\(|\vec{b}_2| = \sqrt{25 + 64 + 16} = \sqrt{105}\)
\( \displaystyle \cos\theta = \frac{20}{14\sqrt{105}} = \frac{10}{7\sqrt{105}} \)
\( \displaystyle \theta = \cos^{-1}\frac{10}{7\sqrt{105}} \)