Vector Algebra

Scalar & Vector Quantities

• Scalar: Has only magnitude (mass, time, speed, work, temperature)
• Vector: Has magnitude and direction and obeys triangle law of addition (displacement, velocity, force)

Representation of Vectors

• Represented by directed line segments
• Magnitude = length of the segment
• Direction = direction from initial to terminal point
• Denoted by a⃗ or AB⃗

Types of Vectors

• Zero / Null vector: Magnitude zero, direction indeterminate
• Unit vector: Magnitude 1, along direction of a⃗ → â = a⃗ / |a⃗|
• î, ĵ, k̂: Unit vectors along x, y, z axes
• Equal vectors: Same magnitude and same direction
• Negative vector: Same magnitude, opposite direction
• Collinear vectors: Parallel to same line
• Like & unlike vectors: Same or opposite direction
• Coplanar vectors: Lie in the same plane
• Position vector: Vector from origin to a point

Vector Addition

• Triangle law: a⃗ + b⃗ = resultant from start of a⃗ to end of b⃗
• Parallelogram law: Diagonal represents sum

Properties

• a⃗ + b⃗ = b⃗ + a⃗ (commutative)
• a⃗ + (b⃗ + c⃗) = (a⃗ + b⃗) + c⃗ (associative)
• a⃗ + 0⃗ = a⃗
• a⃗ + (−a⃗) = 0⃗

Difference of Vectors

• a⃗ − b⃗ = a⃗ + (−b⃗)

Multiplication by a Scalar

• |m a⃗| = |m||a⃗|
• Direction same if m > 0, opposite if m < 0
• 0·a⃗ = 0⃗

Component Form

• a⃗ = x î + y ĵ + z k̂
• |a⃗| = √(x² + y² + z²)
• a⃗ = b⃗ ⇔ x₁=x₂, y₁=y₂, z₁=z₂

Section Formula

• Internal division (m:n): (m b⃗ + n a⃗)/(m+n)
• External division (m:n): (m b⃗ − n a⃗)/(m−n)
• Midpoint: (a⃗ + b⃗)/2

Scalar (Dot) Product

• a⃗ · b⃗ = |a⃗||b⃗| cosθ
• a⃗ · b⃗ = a₁b₁ + a₂b₂ + a₃b₃
• a⃗ ⟂ b⃗ ⇔ a⃗ · b⃗ = 0
• cosθ = (a⃗ · b⃗)/(|a⃗||b⃗|)

Vector (Cross) Product

• a⃗ × b⃗ = |a⃗||b⃗| sinθ n̂
• a⃗ × b⃗ = −(b⃗ × a⃗)
• a⃗ × a⃗ = 0⃗
• a⃗ × b⃗ = | î ĵ k̂ |
  | a₁ a₂ a₃ |
  | b₁ b₂ b₃ |

Area using Cross Product

• Area of parallelogram = |a⃗ × b⃗|
• Area of triangle = ½ |a⃗ × b⃗|

Scalar Triple Product

• [a⃗ b⃗ c⃗] = a⃗ · (b⃗ × c⃗)
• Represents volume of parallelepiped
• [a⃗ b⃗ c⃗] = 0 ⇔ vectors are coplanar
• Volume of tetrahedron = |[a⃗ b⃗ c⃗]| / 6

Vector Triple Product

• a⃗ × (b⃗ × c⃗) = b⃗(a⃗·c⃗) − c⃗(a⃗·b⃗)
• Not commutative
• Result lies in plane of b⃗ and c⃗

Linearly Dependent & Independent Vectors

• Dependent if x₁a⃗₁ + x₂a⃗₂ + … = 0⃗ (not all x = 0)
• Independent if only trivial solution exists
• Two vectors are dependent ⇔ they are parallel

Applications in Geometry

• Line: r⃗ = a⃗ + t b⃗
• Plane: r⃗ = a⃗ + s b⃗ + t c⃗
• Plane (normal form): r⃗ · n⃗ = a⃗ · n⃗
• Distance of point from line or plane using vector formulae

Applications in Mechanics

• Work: W = F⃗ · d⃗
• Moment of force: τ⃗ = r⃗ × F⃗
• Moment of a couple: r⃗ × F⃗

JEE Main Focus Tips

• Component method simplifies most problems
• Check coplanarity using scalar triple product
• Area & volume questions are scoring
• Be careful with direction of cross product