Question

Find the angle between the lines \(\vec r = 3\hat i+2\hat j-4\hat k+\lambda(\hat i+2\hat j+2\hat k)\) and \(\vec r = 5\hat i-2\hat j+\mu(3\hat i+2\hat j+6\hat k)\).

Hint:

Use cosθ = |d1·d2|/(|d1||d2|) with the direction vectors of both lines.

Answer 1

Step 1: Key Formula or Approach:
The angle between two lines depends only on their direction vectors \(\vec d_1\) and \(\vec d_2\), using \(\cos\theta = \dfrac{|\vec d_1\cdot\vec d_2|}{|\vec d_1||\vec d_2|}\).

Step 2: Identifying the direction vectors:
\(\vec d_1 = \hat i+2\hat j+2\hat k\) and \(\vec d_2 = 3\hat i+2\hat j+6\hat k\).

Step 3: Computing the dot product and magnitudes:
\[ \vec d_1\cdot\vec d_2 = (1)(3)+(2)(2)+(2)(6) = 3+4+12 = 19 \]
\[ |\vec d_1| = \sqrt{1+4+4} = 3,\qquad |\vec d_2| = \sqrt{9+4+36} = \sqrt{49} = 7 \]

Step 4: Computing \(\cos\theta\):
\[ \cos\theta = \frac{19}{3\times7} = \frac{19}{21} \]

Final Answer:
The angle between the lines is \(\theta = \cos^{-1}\left(\dfrac{19}{21}\right)\). \[ \boxed{\theta = \cos^{-1}\left(\dfrac{19}{21}\right)} \]