MHT CET202525 Apr 2025Evening ShiftMathematicsThree Dimensional GeometryActual
The lines r =( i + j - k )+ (3 i - j ) and r =(4 i - k )+ (2 i +3 k ) are
Options
- Aintersecting but not perpendicular
- Bperpendicular
- Cparallel
- Dskew lines
Correct answer
A. intersecting but not perpendicular
Step-by-step solution
The lines are defined parametrically as L₁: r = ( i + j - k ) + (3 i - j ) and L₂: r = (4 i - k ) + (2 i + 3 k ) . Parallelism Check: Since b ₁ = 3 i - j and b ₂ = 2 i + 3 k are not scalar multiples, the lines are not parallel. Perpendicularity Check: The dot product b ₁ b ₂ = 6 0 , so the lines are not perpendicular. Intersection Check: Setting the position vectors equal yields the system: 1 + 3 = 4 + 2 1 - = 0 -1 = -1 + 3 Solving gives = 1 and = 0 , which satisfy all equations. The intersection point is r = 4 i -