Question

Find the angle between the lines \( \vec{r} = 3\hat{i} + 8\hat{j} + 3\hat{k} + \mu(\hat{i} + 2\hat{j} - \hat{k}) \) and \( \vec{r} = -3\hat{i} + 9\hat{j} - \hat{k} + \lambda(5\hat{i} + 3\hat{j} + 4\hat{k}) \).

Hint:

Angle calculation only requires the direction vectors (terms following \( \mu \) and \( \lambda \)). The constant vectors represent points on the lines and do not influence the angle.

Answer 1

Step 1: Understanding the Concept:
The angle \( \theta \) between two lines is defined by the angle between their direction vectors \( \vec{b}_1 \) and \( \vec{b}_2 \).
Step 2: Key Formula or Approach:
\[ \cos \theta = \frac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{b}_1||\vec{b}_2|} \] Step 3: Detailed Explanation:
Direction vectors:
\( \vec{b}_1 = \hat{i} + 2\hat{j} - \hat{k} \).
\( \vec{b}_2 = 5\hat{i} + 3\hat{j} + 4\hat{k} \).
Calculate dot product: \( \vec{b}_1 \cdot \vec{b}_2 = (1)(5) + (2)(3) + (-1)(4) = 5 + 6 - 4 = 7 \).
Calculate magnitudes:
\( |\vec{b}_1| = \sqrt{1^2 + 2^2 + (-1)^2} = \sqrt{6} \).
\( |\vec{b}_2| = \sqrt{5^2 + 3^2 + 4^2} = \sqrt{25 + 9 + 16} = \sqrt{50} = 5\sqrt{2} \).
Then:
\[ \cos \theta = \frac{7}{\sqrt{6} \cdot 5\sqrt{2}} = \frac{7}{5\sqrt{12}} = \frac{7}{5 \cdot 2\sqrt{3}} = \frac{7}{10\sqrt{3}} \].
Step 4: Final Answer:
The angle is \( \theta = \cos^{-1}\left(\frac{7}{10\sqrt{3}}\right) \).