Question

The angle between the pair of lines given by \[ \vec{r} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k}) \] and \[ \vec{r} = 5\hat{i} - 2\hat{j} + \mu(3\hat{i} + 2\hat{j} + 6\hat{k}) \] is _____

• A. \( \cos^{-1}\left(\frac{19}{21}\right) \)
• B. \( \sin^{-1}\left(\frac{19}{21}\right) \)
• C. \( \cos^{-1}\left(-\frac{19}{21}\right) \)
• D. \( \cos^{-1}\left(\frac{\sqrt{19}}{21}\right) \)

Hint:

Angle between lines depends only on direction vectors, not position vectors.

Answer 1

The Correct Option is A

Concept: Angle between lines = angle between direction vectors: \[ \cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} \]
Step 1: Direction vectors. \[ \vec{a} = (1,2,2), \quad \vec{b} = (3,2,6) \]
Step 2: Dot product. \[ \vec{a} \cdot \vec{b} = 3 + 4 + 12 = 19 \]
Step 3: Magnitudes. \[ |\vec{a}| = 3, \quad |\vec{b}| = 7 \]
Step 4: \[ \cos\theta = \frac{19}{21} \] \[ \theta = \cos^{-1}\left(\frac{19}{21}\right) \]