The angle between the lines (x - 1)/1 = (y - 1)/1 = (z - 1)/2 and (x -...
Understanding the Lines
To find the angle between the two lines, we first identify their direction ratios from the given equations.
- Line 1: (x - 1)/1 = (y - 1)/1 = (z - 1)/2
- Direction Ratios: (1, 1, 2)
- Line 2: (x - 1)/(-√3 - 1) = (y - 1)/(√3 - 1) = (z - 1)/4
- Direction Ratios: (-√3 - 1, √3 - 1, 4)
Calculating the Angle
The angle θ between two lines with direction ratios (a1, b1, c1) and (a2, b2, c2) can be found using the formula:
cos(θ) = (a1*a2 + b1*b2 + c1*c2) / (√(a1² + b1² + c1²) * √(a2² + b2² + c2²))
- Substituting Values:
- a1 = 1, b1 = 1, c1 = 2
- a2 = -√3 - 1, b2 = √3 - 1, c2 = 4
Calculating Dot Product
- Dot Product:
- (1)(-√3 - 1) + (1)(√3 - 1) + (2)(4)
- = -√3 - 1 + √3 - 1 + 8
- = 6
Calculating Magnitudes
- Magnitude of Line 1:
- √(1² + 1² + 2²) = √6
- Magnitude of Line 2:
- √((-√3 - 1)² + (√3 - 1)² + 4²)
- = √(4 + 4√3) = 8
Final Calculation
Now, substituting back into the cosine formula:
cos(θ) = 6 / (√6 * 8)
This simplifies to cos(θ) = 1/2, leading to:
- Angle θ = π/3 (60 degrees)
Hence, the angle between the lines is π/3, which corresponds to option 'C'.