Explanation:

Let’s solve it step-wise:

- Direction ratios of lines:

Line 1: 1, 1, -2

Line 2: (v3 – 1), (–v3 – 1), –4

- Angle ? between lines:

cos? = (a1a2 + b1b2 + c1c2)/[v(a1² + b1² + c1²) × v(a2² + b2² + c2²)]

- Numerator:

= 1×(v3–1) + 1×(–v3–1) + (–2)×(–4)

= (v3–1) + (–v3–1) + 8

= (v3–v3) + (–1–1) + 8

= 0 –2 + 8 = 6

- Denominator:

= v(1²+1²+4) × v((v3–1)² + (–v3–1)² + 16)

= v6 × v[(3–2v3+1)+(3+2v3+1)+16]

= v6 × v[(4–2v3)+(4+2v3)+16]

= v6 × v[8+16] = v6 × v24 = v6 × 2v6 = 12

- cos? = 6/12 = 1/2 ? ? = p/3

Correct Option:

- Option 1: p/3

- Options explained:

– Option 2: p (180°) is a straight angle

– Option 3: p/6 (30°), not matching calculation

– Option 4: p/2 (90°), right angle, not matching

- Option 1 (p/3 or 60°) is correct.