Vectors & 3D Geometry for JEE Maths
Vectors and 3D Geometry is a Class 12 topic that contributes 2-3 questions per JEE paper. It's considered one of the most scoring chapters because the problems follow clear patterns and formulas are straightforward to apply. If you learn the formulas and practice the standard problem types, this is almost guaranteed marks.
Prerequisites: Straight Lines, Basic Coordinate Geometry
What JEE Tests
Key areas:
- Dot product: angle between vectors, projection, work done
- Cross product: area of parallelogram/triangle, perpendicular vector
- Scalar triple product: volume of parallelepiped, coplanarity test
- Equation of a line in 3D (vector and Cartesian forms)
- Equation of a plane (normal form, intercept form, three-point form)
- Angle between line and plane, two planes, two lines
- Shortest distance between two skew lines
- Image of a point in a plane, foot of perpendicular
Vector Operations
Dot product: ⃗a · ⃗b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃. Result is a scalar. Use for: finding angles, projections, and checking perpendicularity (⃗a · ⃗b = 0).
Cross product: ⃗a × ⃗b = |a||b|sinθ ñ, where ñ is the unit vector perpendicular to both. Computed using the determinant of [î ĵ k̂; a₁ a₂ a₃; b₁ b₂ b₃]. Use for: areas, finding perpendicular directions, and checking parallelism (⃗a × ⃗b = 0).
Scalar triple product: [⃗a ⃗b ⃗c] = ⃗a · (⃗b × ⃗c). Equals zero if vectors are coplanar. Its absolute value = volume of parallelepiped.
3D Geometry: Lines and Planes
Line through point ⃗a in direction ⃗b: ⃗r = ⃗a + λ⃗b. In Cartesian: (x-a₁)/b₁ = (y-a₂)/b₂ = (z-a₃)/b₃.
Plane with normal ⃗n through point ⃗a: ⃗r · ⃗n = ⃗a · ⃗n. In Cartesian: ax + by + cz = d.
Shortest distance between skew lines ⃗r = ⃗a₁ + λ⃗b₁ and ⃗r = ⃗a₂ + μ⃗b₂: d = |(⃗a₂ - ⃗a₁) · (⃗b₁ × ⃗b₂)| / |⃗b₁ × ⃗b₂|. This is a very common JEE question.
Key Formulas
Common Mistakes to Avoid
- Mistake: Confusing dot product (scalar result) with cross product (vector result)
- Mistake: Wrong order in cross product (a×b = -b×a)
- Mistake: Forgetting to normalize the direction vector when needed
- Mistake: Using 2D distance formula in 3D problems
How to Practice This Topic
This is a formula-heavy chapter. First memorize all formulas (2 days). Then solve 10 dot/cross product problems, 10 line/plane equation problems, and 5 shortest distance problems. Scoring topic — don't skip it.
Formula Sheets
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