By Abhishek Jaiswal, Mathematics faculty at Pramana Institute (IIT Roorkee, 18 years teaching JEE)
Vectors is one of the most reliable chapters in JEE Main: published counts put it at about two questions a year, more than almost any other Maths chapter. The questions are rarely hard. They test whether you pick the right product (dot, cross or triple) and apply it without a sign error. Learn the formulas below as a set, and this chapter becomes some of the fastest marks in the paper.
Basics
| Quantity | Formula (for a = a₁i + a₂j + a₃k) |
|---|---|
| Magnitude | |a| = √(a₁² + a₂² + a₃²) |
| Unit vector | â = a/|a| |
| Direction cosines | l = a₁/|a|, m = a₂/|a|, n = a₃/|a|, and l² + m² + n² = 1 |
| Section formula (internal, m : n) | r = (mb + na)/(m + n) |
| Mid-point | r = (a + b)/2 |
| Collinear points A, B, C | AB = λ AC for some scalar λ |
Dot product
| Use | Formula |
|---|---|
| Definition | a · b = |a| |b| cos θ = a₁b₁ + a₂b₂ + a₃b₃ |
| Angle between vectors | cos θ = (a · b)/(|a| |b|) |
| Perpendicular vectors | a · b = 0 |
| Projection of a on b | (a · b)/|b| |
| Work done | W = F · d |
Cross product
| Use | Formula |
|---|---|
| Magnitude | |a × b| = |a| |b| sin θ |
| Order matters | a × b = −(b × a) |
| Parallel vectors | a × b = 0 |
| Unit vector perpendicular to a and b | ±(a × b)/|a × b| |
| Area of parallelogram (sides a, b) | |a × b| |
| Area of parallelogram (diagonals d₁, d₂) | ½ |d₁ × d₂| |
| Area of triangle | ½ |a × b| |
| Lagrange's identity | |a × b|² = |a|²|b|² − (a · b)² |
a × b = determinant with rows (î, ĵ, k̂), (a₁, a₂, a₃), (b₁, b₂, b₃) = (a₂b₃ − a₃b₂) î − (a₁b₃ − a₃b₁) ĵ + (a₁b₂ − a₂b₁) k̂
Triple products
| Use | Formula |
|---|---|
| Scalar triple product | [a b c] = a · (b × c), the determinant of the three rows of components |
| Cyclic rule | [a b c] = [b c a] = [c a b] = −[b a c] |
| Repeated vector | [a a b] = 0 |
| Volume of parallelepiped | |[a b c]| |
| Volume of tetrahedron | (1/6) |[a b c]| |
| Coplanar vectors | [a b c] = 0 |
| Vector triple product | a × (b × c) = (a · c) b − (a · b) c |
Worked example
For a = i + 2j + 3k and b = 3i − j + 2k, find the area of the parallelogram they form.
a × b = i(2·2 − 3·(−1)) − j(1·2 − 3·3) + k(1·(−1) − 2·3) = 7i + 7j − 7k.
Area = |a × b| = 7√3 square units. Check with Lagrange's identity: |a|² = 14, |b|² = 14 and a · b = 3 − 2 + 6 = 7, so 14 × 14 − 49 = 147 = (7√3)².
Three mistakes that cost marks
- Moving the brackets in a vector triple product. a × (b × c) is not equal to (a × b) × c. Expand each one with its own formula.
- Mixing up projection and component. The projection of a on b is a number, (a · b)/|b|. The component vector is that number times the unit vector b̂.
- Forgetting the ±. A question asking for "a unit vector perpendicular to both" has two answers, ±(a × b)/|a × b|.
FAQs
How many questions come from vectors in JEE Main? Usually one or two, often combined with 3D geometry.
When do I use the scalar triple product? For volumes and for checking whether three vectors (or four points) are coplanar.
Are triple products in the Class 12 CBSE syllabus? The current CBSE syllabus focuses on dot and cross products, so check the latest CBSE syllabus for your board exam. JEE Main uses both triple products, so learn them for JEE.
Practise vectors with weekly tests in a batch of at most 40 at Pramana, Geeta Bhawan Square, Indore. See our IIT JEE program or book free counselling.
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