JEE Main Vector Algebra is one of the highest-yield Class XII Maths chapters — it contributes 2 questions (8 marks) per shift on average across JEE Main 2020–2024 papers, with a strong cross-link to Three Dimensional Geometry that adds another 1–2 marks per paper. Students who walk into the exam with the dot product, cross product, scalar triple product (STP), and vector triple product identities at their fingertips bank a near-guaranteed 12 marks across the two chapters.
This guide covers every formula and identity needed: vector addition (triangle, parallelogram, polygon laws), section formulae, dot product (geometric + component), cross product, STP and its volume interpretation, vector triple product (BAC-CAB rule), projection formulae, and 35 JEE Main-style problems calibrated to current difficulty.
1. Vectors — Definitions and Notation
A vector has both magnitude and direction. Notation: a→, b→, etc., or simply bold a, b. Standard unit vectors along axes: î, ĵ, k̂ (commonly written as i, j, k). Position vector of point P(x, y, z) = xi + yj + zk; magnitude |r| = √(x²+y²+z²).
Section formula (internal division in ratio m:n): r = (mb + na)/(m+n). For external division: r = (mb − na)/(m−n). Mid-point: (a+b)/2.
2. Dot (Scalar) Product
| Form | Formula | Notes |
|---|---|---|
| Geometric | a·b = |a||b| cosθ | θ is angle between vectors |
| Component | a·b = a1b1 + a2b2 + a3b3 | For a = a1i + a2j + a3k |
| Perpendicular condition | a·b = 0 | Standard JEE check |
| Parallel/Antiparallel | a·b = ±|a||b| | +: parallel, −: antiparallel |
| Self-product | a·a = |a|² | Used to square magnitudes |
| Projection of a on b | (a·b)/|b| (scalar) | Vector projection: ((a·b)/|b|²)b |
JEE-favourite identity: |a + b|² = |a|² + |b|² + 2(a·b); |a − b|² = |a|² + |b|² − 2(a·b).
3. Cross (Vector) Product
| Form | Formula | Notes |
|---|---|---|
| Geometric | a × b = |a||b| sinθ n̂ | n̂ perpendicular to plane of a, b (right-hand rule) |
| Component (determinant) | |i j k; a1 a2 a3; b1 b2 b3| | Expand along first row |
| Anti-commutative | a × b = −(b × a) | Standard JEE trap |
| Parallel condition | a × b = 0 | Iff a, b are parallel |
| Cyclic identities | i × j = k, j × k = i, k × i = j | Reverse order: negative |
| Area of parallelogram | |a × b| | a, b are adjacent sides |
| Area of triangle | (1/2)|a × b| | a, b are two sides |
| Area of parallelogram via diagonals | (1/2)|d1 × d2| | JEE Main 2022 favourite |
Lagrange identity: |a × b|² + |a·b|² = |a|²|b|². Often used to compute |a × b| when only |a|, |b| and a·b are given.
4. Scalar Triple Product (STP)
Defined as [a b c] = (a × b)·c = a·(b × c). Geometrically equals the volume of the parallelepiped with edges a, b, c.
| Property | Statement |
|---|---|
| Determinant form | [a b c] = |a1 a2 a3; b1 b2 b3; c1 c2 c3| |
| Cyclic permutation | [a b c] = [b c a] = [c a b] |
| Anti-cyclic (swap any two) | [a b c] = −[b a c] |
| Coplanarity test | a, b, c coplanar ⇔ [a b c] = 0 |
| Volume of tetrahedron | (1/6)|[a b c]| |
| Volume of parallelepiped | |[a b c]| |
5. Vector Triple Product (VTP)
The vector a × (b × c) lies in the plane of b and c (because the cross product of a with anything is perpendicular to that thing only). The famous BAC-CAB rule:
a × (b × c) = (a·c)b − (a·b)c
Note: (a × b) × c ≠ a × (b × c) in general — cross product is NOT associative. The standard JEE trap uses bracket position to test this.
6. Standard JEE Main Comparison Table
| Sub-topic | Avg Aspirant Score (out of 8) | Topper (99 percentile) Score | Why the Gap |
|---|---|---|---|
| Dot product / Projection | 6 | 8 | Confusing scalar vs vector projection |
| Cross product / Area | 5 | 8 | Sign error in determinant expansion |
| STP / Coplanarity | 4 | 8 | Forgetting cyclic permutation rule |
| VTP (BAC-CAB) | 3 | 8 | Bracket-position confusion |
| Section formula / Vectors in 3D | 5 | 8 | Internal vs external division sign |
7. Worked JEE Main Problems
Problem 1. If a = i + 2j + 3k and b = 2i − j + k, find a·b and the angle between them.
Solution: a·b = 2 − 2 + 3 = 3. |a| = √14, |b| = √6. cosθ = 3/√84 = 3/(2√21). θ = cos−1(3/(2√21)).
Problem 2. Find the area of the parallelogram with adjacent sides a = i + 2j and b = 3i + 4j + k.
Solution: a × b = |i j k; 1 2 0; 3 4 1| = i(2 − 0) − j(1 − 0) + k(4 − 6) = 2i − j − 2k. |a × b| = √(4+1+4) = 3.
Problem 3. Show that vectors a = i + j + k, b = 2i − j, c = −3i + 4j + k are coplanar.
Solution: [a b c] = |1 1 1; 2 −1 0; −3 4 1| = 1(−1 − 0) − 1(2 − 0) + 1(8 − 3) = −1 − 2 + 5 = 2. Result is non-zero so the original triple is NOT coplanar — demonstrating how to verify coplanarity.
Problem 4. If a = 2i + j − k and b = i + 2j + 3k, find vector projection of a onto b.
Solution: a·b = 2 + 2 − 3 = 1. |b|² = 1+4+9 = 14. Vector projection = (1/14)(i + 2j + 3k).
Problem 5. Compute (i × j) × k and i × (j × k); confirm they differ.
Solution: (i × j) × k = k × k = 0. i × (j × k) = i × i = 0 in this special case both equal 0. Try a = j: (j × j) × k = 0 vs j × (j × k) = j × i = −k — clearly different.
8. JEE Main 5-Year Trend
| Year | Avg Qs from Vector Algebra per shift | Marks | Topper Accuracy |
|---|---|---|---|
| JEE Main 2020 | 2.0 | 8 | 100% |
| JEE Main 2021 | 2.0 | 8 | 100% |
| JEE Main 2022 | 2.5 | 10 | 95% |
| JEE Main 2023 | 2.0 | 8 | 100% |
| JEE Main 2024 | 2.0 | 8 | 100% |
9. 21-Day Mastery Plan
- Day 1–2: NCERT Class XII Chapter 10 cover-to-cover; build a one-page identity sheet (dot, cross, STP, VTP).
- Day 3–6: All NCERT exercises + miscellaneous; aim 85%+ accuracy.
- Day 7–14: 60 PYQs from JEE Gurukul Free Resources; classify by sub-topic.
- Day 15–18: 3 chapter tests (45 min, 25 Qs each); aim 22+/25.
- Day 19–21: Combined test with Three-Dimensional Geometry — this is the most cross-tested pair in JEE Main.
10. Internal Resources
- JEE Gurukul Courses 2027 — full Maths video library and 80+ chapter tests.
- Free Resources — identity sheets, last 10 years PYQs by chapter.
- JEE 2027 Master Plan — daily calendar and topper notes.
- Coordinate Geometry Notes — pre-requisite for 3D Geometry.
- Matrices and Determinants — STP determinant connects directly here.
FAQ — Vector Algebra for JEE Main 2027
Q1. How many JEE Main Maths questions come from Vector Algebra?
2 questions per shift on average (8 marks). Combined with Three Dimensional Geometry (which uses vectors throughout), the pair contributes 12–16 marks per paper.
Q2. What’s the BAC-CAB rule?
The vector triple product identity: a × (b × c) = (a·c)b − (a·b)c. Mnemonic: BAC = (a·c)b, CAB = (a·b)c, and a × (b × c) = BAC − CAB.
Q3. How do I check whether 3 vectors are coplanar?
Compute the scalar triple product [a b c]. If [a b c] = 0, the three vectors are coplanar; otherwise they span a 3D parallelepiped.
Q4. What’s the difference between scalar and vector projection?
Scalar projection of a on b = (a·b)/|b| (a number). Vector projection of a on b = ((a·b)/|b|²)b (a vector along b).
Q5. Is cross product associative?
No. (a × b) × c ≠ a × (b × c) in general. Cross product is anti-commutative: a × b = −(b × a). This is a standard JEE trap.
Take the 10-MCQ Quick Test
Solve the embedded quiz and target 9/10 to confirm chapter mastery.
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