JEE Main Vectors and 3D Geometry: Complete Formula and Problem Guide
Vectors and 3D Geometry contribute 2-3 questions in JEE Main Mathematics. With consistent formulas and clear visualization skills, this can be a reliable scoring topic.
Vectors: Dot and Cross Products
Dot product: a·b = |a||b|cosθ. Cross product: |a×b| = |a||b|sinθ. Applications: angle between vectors, projection, area of parallelogram (|a×b|/2). Unit vector formula: a/|a|. Condition for perpendicular vectors: a·b = 0.
Vector Triple Products
Scalar triple product [a b c] = a·(b×c) = volume of parallelepiped. Coplanar condition: [a b c] = 0. Vector triple product: a×(b×c) = (a·c)b - (a·b)c. These appear in 1-2 JEE Main problems.
3D Geometry: Lines
Equation of line through point (x₁,y₁,z₁) with direction (a,b,c): (x-x₁)/a = (y-y₁)/b = (z-z₁)/c. Distance between two skew lines: |[(a₂-a₁) (b₁ b₂)]| / |b₁×b₂|. Angle between two lines: cosθ = |b₁·b₂|/(|b₁||b₂|).
3D Geometry: Planes
Equation ax + by + cz = d. Normal vector = (a,b,c). Distance from point to plane: |ax₁+by₁+cz₁-d|/√(a²+b²+c²). Equation of plane through three points. Angle between two planes = angle between normals.
JEE Main Question Patterns
Most common: find angle between line and plane, find distance from point to plane, determine if lines are coplanar and find their intersection. Practice these 3 question types repeatedly — they cover 90% of 3D Geometry JEE Main questions.
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