Question

If \(θ\) is the angle between the lines \(\frac{x-1}{2} = \frac{2y+3}{4}; z = -2\) and \(x = 1; \frac{y-1}{2} = \frac{z+1}{2}\), then

• A. \(θ = \frac{π}{6}\)
• B. \(θ = \frac{π}{3}\)
• C. \(θ = \frac{π}{4}\)
• D. \(θ = \frac{π}{2}\)

Hint:

Find the direction ratios and use the dot product formula.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Rewrite each line in standard form to read its direction ratios.

Step 2: Key Formula or Approach
Line 1: \(\dfrac{x-1}{2}=\dfrac{y+3/2}{2}\), \(z=-2\), direction \((2,2,0)\sim(1,1,0)\). Line 2: \(x=1\), \(\dfrac{y-1}{2}=\dfrac{z+1}{2}\), direction \((0,2,2)\sim(0,1,1)\).

Step 3: Detailed Explanation
\[ \cos\theta=\frac{|1\cdot0+1\cdot1+0\cdot1|}{\sqrt2\cdot\sqrt2}=\frac12 \]
So \(\theta=\dfrac\pi3\).

Final Answer:
The angle is \(\pi/3\), option (B). \[ \boxed{\dfrac\pi3\ \text{(B)}} \]