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Two lines r = a₁ + λb₁ and r = a₂ + μb₂ are coplanar if and only if
A(a₂ - a₁) · (b₁ × b₂) = 0
B(a₂ - a₁) × (b₁ + b₂) = 0
Cb₁ · b₂ = 0
Db₁ × b₂ = 0
Answer & Solution
Correct answer: A. (a₂ - a₁) · (b₁ × b₂) = 0
1. Two lines are coplanar when the vector joining a point on one to a point on the other lies in the plane spanned by their directions.
2. That plane has normal b₁ × b₂.
3. The connecting vector (a₂ - a₁) must be perpendicular to this normal.
4. The condition is (a₂ - a₁) · (b₁ × b₂) = 0.
_Source: NCERT Class 12 Maths Part 2 Ch 11 "Three Dimensional Geometry", §11.7_
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