Question

The measure of the angle between the lines $x = k - 1, y = 2k + 1, z = 2k + 3, k \in \mathbb{R}$ and $\frac{x+1}{2} = \frac{y-2}{1} = \frac{z-1}{2}$ is

• A. $\cos^{-1}\left(\frac{2}{3}\right)$
• B. $\cos^{-1}\left(\frac{8}{9}\right)$
• C. $\cos^{-1}\left(\frac{5}{12}\right)$
• D. $\sin^{-1}\left(\frac{8}{9}\right)$

Hint:

Always convert parametric line equations into symmetric (Cartesian) form to reliably read off the direction ratios from the denominators. Ensure the coefficients of $x, y, z$ in the numerators are all $1$ before reading the denominators.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To find the angle between two lines in 3D space, we need to extract their direction ratios and apply the angle formula based on the dot product of their direction vectors.

Step 2: Key Formula or Approach:
For lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$, the angle $\theta$ between them is given by: \[ \cos\theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \] First, convert the parametric equations of the first line into symmetric form to easily identify its direction ratios.

Step 3: Detailed Explanation:
Let's analyze the first line, given in parametric form: $x = k - 1 \implies x + 1 = k \implies \frac{x + 1}{1} = k$ $y = 2k + 1 \implies y - 1 = 2k \implies \frac{y - 1}{2} = k$ $z = 2k + 3 \implies z - 3 = 2k \implies \frac{z - 3}{2} = k$ Combining these, the symmetric equation for the first line is: \[ \frac{x + 1}{1} = \frac{y - 1}{2} = \frac{z - 3}{2} \] The direction ratios of the first line are $\vec{d}_1 = (1, 2, 2)$. The second line is already in symmetric form: \[ \frac{x + 1}{2} = \frac{y - 2}{1} = \frac{z - 1}{2} \] The direction ratios of the second line are $\vec{d}_2 = (2, 1, 2)$. Now, apply the cosine formula for the angle $\theta$: \[ \cos\theta = \frac{|(1)(2) + (2)(1) + (2)(2)|}{\sqrt{1^2 + 2^2 + 2^2} \sqrt{2^2 + 1^2 + 2^2}} \] \[ \cos\theta = \frac{|2 + 2 + 4|}{\sqrt{1 + 4 + 4} \sqrt{4 + 1 + 4}} \] \[ \cos\theta = \frac{8}{\sqrt{9} \sqrt{9}} \] \[ \cos\theta = \frac{8}{3 \times 3} = \frac{8}{9} \] Therefore, the angle is $\theta = \cos^{-1}\left(\frac{8}{9}\right)$.

Step 4: Final Answer:
The measure of the angle is $\cos^{-1}\left(\frac{8}{9}\right)$.