Question

(b)(i) Find the angle between the pair of lines: \[ \frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4}, \quad \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2}. \]

Hint:

The dot product helps calculate angles between vectors effectively.

Answer 1

For two lines: \[ \vec{b}_1 = (3, 5, 4), \quad \vec{b}_2 = (1, 1, 2). \] The angle \( \theta \) is given by: \[ \cos \theta = \frac{\vec{b}_1 \cdot \vec{b}_2}{|\vec{b}_1||\vec{b}_2|}. \] Substitute \( \vec{b}_1 \cdot \vec{b}_2 = 3 \cdot 1 + 5 \cdot 1 + 4 \cdot 2 = 16 \), \( |\vec{b}_1| = \sqrt{3^2 + 5^2 + 4^2} = \sqrt{50} \), \( |\vec{b}_2| = \sqrt{1^2 + 1^2 + 2^2} = \sqrt{6} \). Then: \[ \cos \theta = \frac{16}{\sqrt{50} \cdot \sqrt{6}}. \]