COMEDK2024MathematicsThree Dimensional GeometryActual
The lines r =(2 -3 k )+ ( +2 +3 k ) and r =(2 +6 +3 k )+ (2 +3 +4 k ) are
Options
- AParallel lines
- BIntersecting lines
- CSkew lines
- Dco-incident lines
Correct answer
B. Intersecting lines
Step-by-step solution
The given lines are r = a ₁ + b ₁ and r = a ₂ + b ₂ , where a ₁ = 2 j - 3 k , b ₁ = i + 2 j + 3 k , a ₂ = 2 i + 6 j + 3 k , and b ₂ = 2 i + 3 j + 4 k . First, check if the lines are parallel. Since b ₁ b ₂ = ( i + 2 j + 3 k ) (2 i + 3 j + 4 k ) = vmatrix i & j & k 1 & 2 & 3 2 & 3 & 4 vmatrix = i (8-9) - j (4-6) + k (3-4) = - i + 2 j - k 0 , the lines are not parallel. Next, check if the lines intersect by calculating the scalar triple product ( a ₂ - a ₁) ( b ₁ b ₂) . a ₂ - a ₁ = (2 i + 6 j + 3 k ) - (2 j - 3 k ) =