Question

The cosine of the angle included between the lines \[ \vec{r_1} = (2i + j - 2k) + \lambda (i - 2j - 2k) \quad \text{and} \quad \vec{r_2} = (i + j + 3k) + \mu (3i + 2j - 6k) \] where \( \lambda, \mu \in \mathbb{R} \) is

• A. \( \frac{13}{21} \)
• B. \( \frac{11}{21} \)
• C. \( \frac{3}{21} \)
• D. \( \frac{17}{21} \)

Hint:

To find the cosine of the angle between two lines, use the formula \( \cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} \), where \( \vec{a} \) and \( \vec{b} \) are the direction ratios of the lines.

Answer 1

The Correct Option is B

Step 1: Understand the given lines.
The lines are given in parametric form. To find the cosine of the angle between the two lines, we need to compute the direction ratios (vectors) of the lines, which are \( \vec{d_1} = (1, -2, -2) \) and \( \vec{d_2} = (3, 2, -6) \), corresponding to the coefficients of \( \lambda \) and \( \mu \) in the parametric equations.

Step 2: Use the formula for cosine of the angle.
The formula for the cosine of the angle \( \theta \) between two vectors \( \vec{a} \) and \( \vec{b} \) is: \[ \cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} \] Substitute \( \vec{a} = \vec{d_1} \) and \( \vec{b} = \vec{d_2} \) into the formula: \[ \cos \theta = \frac{(1)(3) + (-2)(2) + (-2)(-6)}{\sqrt{1^2 + (-2)^2 + (-2)^2} \cdot \sqrt{3^2 + 2^2 + (-6)^2}} \] Simplifying, we find \( \cos \theta = \frac{11}{21} \).

Step 3: Conclusion.
Thus, the cosine of the angle between the lines is \( \frac{11}{21} \), corresponding to option (B).