Vector Algebra

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Vector Algebra — Key Facts for JEE Main Vector: has magnitude and direction; written as →a or a Unit vector: vector with magnitude 1; ā = →a/|→a| Dot product: →a · →b = |→a||→b| cos θ; also = a₁b₁ + a₂b₂ + a₃b₃ in components Cross product: →a × →b = |→a||→b| sin θ n̂ where n̂ is perpendicular to both (right-hand rule) ⚡ Exam tip: JEE Main tests dot and cross product properties, projection, and vector equations — projection formula is a favourite!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Vector Algebra — JEE Main Study Guide

Vector representation: In 3D: →a = a₁î + a₂ĵ + a₃k̂; magnitude |→a| = √(a₁² + a₂² + a₃²) If →a makes angles α, β, γ with x, y, z axes: a₁ = |→a| cos α, a₂ = |→a| cos β, a₃ = |→a| cos γ Direction cosines: cos²α + cos²β + cos²γ = 1

Equality: Two vectors are equal if they have same magnitude and direction (parallel and same length)

Addition: →a + →b = (a₁+b₁)î + (a₂+b₂)ĵ + (a₃+b₃)k̂ Triangle law: →a and →b placed head-to-tail; parallelogram law same result

Scalar (dot) product: →a · →b = |→a||→b| cos θ = a₁b₁ + a₂b₂ + a₃b₃ Properties: →a · →a = |→a|²; →a · →b = →b · →a (commutative); →a · (→b + →c) = →a · →b + →a · →c Perpendicular: →a · →b = 0 iff →a ⊥ →b Parallel: →a · →b = |→a||→b| when same direction; = −|→a||→b| when opposite

Vector (cross) product: →a × →b = |→a||→b| sin θ n̂ (right-hand rule) →a × →b = (a₂b₃ − a₃b₂)î − (a₁b₃ − a₃b₁)ĵ + (a₁b₂ − a₂b₁)k̂ (determinant form) Properties: →a × →b = −(→b × →a) (anti-commutative); →a × →a = →0; →a × (→b + →c) = →a × →b + →a × →c Parallel: →a × →b = →0 iff →a ∥ →b

Projection: Projection of →a on →b = (→a · →b̂) = (→a · →b)/|→b| Scalar projection = (→a · →b)/|→b|; Vector projection = [(→a · →b)/|→b|²] →b

Section formula: Point dividing AB in ratio m:n (internal): →r = (m→b + n→a)/(m+n) External division: →r = (m→b − n→a)/(m−n)

Collinearity: Three points A, B, C are collinear if →AB × →AC = →0

Coplanarity: Four points are coplanar if (→B−→A) · [(→C−→A) × (→D−→A)] = 0

• Key formula: →a · →b = |→a||→b| cos θ; →a × →b = |→a||→b| sin θ n̂; projection = (→a · →b)/|→b|
• Common trap: →a × →b = −(→b × →a) — cross product is not commutative! Also, →a · →b = 0 doesn’t mean one vector is zero (unless magnitude is zero)
• Exam weight: 1–2 questions per year (4–8 marks); frequently combined with 3D geometry

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Vector Algebra — Comprehensive JEE Main Notes

Triple products:

• Scalar triple product: [→a →b →c] = →a · (→b × →c) = determinant of components
• Volume of parallelepiped = |→a · (→b × →c)| = |[→a →b →c]|
• Vector triple product: →a × (→b × →c) = →b(→a · →c) − →c(→a · →b) (BAC-CAB rule)

Work done: W = →F · →d = |→F||→d| cos θ

Moment of force: Moment about point O = →r × →F (where →r is position vector of point of application)

Application in physics:

• Angular momentum: →L = →r × →p = →r × m→v
• Torque: →τ = →r × →F
• Area of parallelogram: |→a × →b|
• Area of triangle: ½|→a × →b|

Lagrange’s identity: |→a × →b|² = |→a|²|→b|² − (→a · →b)²

Coplanar vectors: Three vectors are coplanar if scalar triple product is zero: →a · (→b × →c) = 0

Linear combination: If →r = α→a + β→b + γ→c and vectors are non-coplanar, then α, β, γ are unique

Reciprocal system: For three non-coplanar vectors →a, →b, →c: →a’ = (→b × →c)/(→a · (→b × →c)); similarly for →b’, →c’ Property: →a · →a’ = 1; →a · →b’ = 0 for b’ ≠ a’

Vector equation of line: Through point →a with direction →b: →r = →a + t→b (parametric) Two-point form: →r = →a + t(→b − →a)

Vector equation of plane: Through point →a with normal →n: →n · (→r − →a) = 0 Through three points: using cross product to find normal

Perpendicular distance from point to line: d = |(→b − →a) × (→c − →a)| / |→b − →a| where line passes through →a, →b and point is →c

Perpendicular distance from point to plane: d = |→n · (→r₀ − →a)| / |→n| where plane is →n · (→r − →a) = 0 and point is →r₀

Transformation between coordinate systems: If vectors are expressed in terms of unit vectors: →a = a₁î + a₂ĵ + a₃k̂, these are the components

Using vector methods for geometry:

• Perpendicular bisector of AB: set of points P where →PA · →AB = 0
• Angle bisector: points where projection on two direction vectors are equal

Important identities:

• (→a + →b)² = →a² + →b² + 2→a · →b (note: →a² means →a · →a)

• (→a − →b)² = →a² + →b² − 2→a · →b

• (→a + →b) × (→a − →b) = 2(→b × →a)

• Remember: →a · →b = a₁b₁ + a₂b₂ + a₃b₃; →a × →b = determinant; scalar triple product gives volume; projection formula: (→a · →b̂) = (→a · →b)/|→b|

• Previous years: “Find angle between vectors →a = î + ĵ and →b = ĵ + k̂” [2023]; “Find projection of →a on →b if →a = 2î + 4ĵ − k̂ and →b = 3î + 2ĵ + 2k̂” [2024]; “Find unit vector perpendicular to both →a = î + ĵ and →b = ĵ + k̂” [2024]

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📊 JEE Main Exam Essentials

Detail	Value
Questions	90 (30 per subject)
Time	3 hours
Marks	300 (90 per subject)
Section	Physics (30), Chemistry (30), Mathematics (30)
Negative	−1 for wrong answer
Mode	Computer-based

🎯 High-Yield Topics for JEE Main Mathematics

• Calculus (Differentiation + Integration) — ~35 marks combined
• Coordinate Geometry (straight lines, circles, conics) — ~20 marks
• Algebra (Complex Numbers, Quadratics, P&C, Probability) — ~25 marks
• Trigonometry + Inverse Trigonometry — ~15 marks
• Vector + 3D — ~15 marks

📝 Previous Year Question Patterns

• Vector Algebra: 1–2 questions per year, 4–8 marks
• Common patterns: dot and cross product calculations, projection, unit vectors, angle between vectors
• Weight: medium frequency, high scoring if formulas are known

💡 Pro Tips

• Always use the determinant formula for cross product — it’s the safest approach in 3D
• For angle between vectors, use cos θ = (→a · →b)/(|→a||→b|) — but first check they aren’t zero vectors
• Projection of →a on →b is scalar = (→a · →b)/|→b|; vector projection = [(→a · →b)/|→b|²] →b
• Remember: →a × →b is perpendicular to both →a and →b (right-hand rule for direction)
• For coplanarity of 4 points, compute scalar triple product of three vectors from one point
• When finding unit vector in direction of →a, it’s →a/|→a| — but only if →a is not zero

🔗 Official Resources

• NTA Official JEE Main
• JEE Main Syllabus PDF

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