Basics
Scalar: Only magnitude (e.g., mass, temperature)
Vector: Magnitude + Direction (e.g., velocity, force)
Vector Addition: Triangle / Parallelogram / Polygon Law
- Commutative: a + b = b + a
- Associative: a + (b + c) = (a + b) + c
- Triangle inequality: |a − b| ≤ |a + b| ≤ |a| + |b|
- Two non-zero non-collinear vectors define a plane
Section Formula
Point dividing join of p and q in ratio m:n (internal)
r = (n p + m q) / (m + n) External: r = (n p − m q) / (n − m)
r = (n p + m q) / (m + n) External: r = (n p − m q) / (n − m)
Vector (Cross) Product a × b
|a × b| = |a||b| sinθ Direction: Right-hand rule
Area of parallelogram = |a × b|
Area of triangle = (1/2)|a × b|
Area of parallelogram = |a × b|
Area of triangle = (1/2)|a × b|
| Property | Result |
|---|---|
| a × a = 0 | Yes |
| a × b = − (b × a) | Anti-commutative |
| λa × μb = λμ (a × b) | Yes |
| a × (b + c) = a × b + a × c | Distributive |
| a × b = 0 | a ∥ b (or one is zero) |
i × j = k j × k = i k × i = j
j × i = −k etc.
j × i = −k etc.
a = a₁i + a₂j + a₃k b = b₁i + b₂j + b₃k
a × b = | i j k |
| a₁ a₂ a₃ |
| b₁ b₂ b₃ | = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k
a × b = | i j k |
| a₁ a₂ a₃ |
| b₁ b₂ b₃ | = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k
Unit vector ⊥ to both a and b: ± (a × b) / |a × b|
Scalar (Dot) Product a · b
a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃
Projection of a on b = (a · b)/|b|
- a · b = 0 ⇔ a ⊥ b
- i · i = j · j = k · k = 1
- i · j = j · k = k · i = 0
- a · a = |a|²
Scalar Triple Product [a b c] = a · (b × c)
Volume of parallelepiped = |[a b c]|
- [a b c] = [b c a] = [c a b]
- [a b c] = −[a c b]
- [ka b c] = k [a b c]
- [a b c] = 0 ⇔ a, b, c coplanar
- Can interchange dot & cross: [a b c] = (a × b) · c = a · (b × c)
Vector Triple Product a × (b × c)
a × (b × c) = (a · c)b − (a · b)c
Lies in plane of b and c (not generally equal to (a × b) × c)
JEE Must-Know
High Weightage Topics:
Section formula (internal/external)
Cross product magnitude & direction
Area/Volume using vector products
Scalar triple product = 0 for coplanarity
Vector triple product expansion
Shortest distance, angle between lines/planes (using vectors)