Ch. 8 Vectors - Precalculus

Parallel Vectors

Parallel Vectors

• Same/opposite direction

• Not necessarily same magnitude

Equivalent vectors

• Same magnitude

• Same direction

Opposite vectors

• Same magnitude

• Opposite direction (-a)

Resultant vector

• The sum of 2 or more vectors

Zero vector

• AKA null vector

• Magnitude of 0

• No specific direction

• Resultant of 2 opposite vectors

Multiplying Vectors with Scalars

• Vector: v

• Real number scalar: k

• Scalar multiple: kv

  • Magnitude of |k| |v|

  • Direction based on the sign of k

    • k > 0 ~ same direction as v

    • k < 0 ~ opposite direction as v

Component Form

• ⟨Δx, Δy⟩

  • ⟨x₂ - x₁ y₂ - y₁⟩

Vector Operations

• a + b = ⟨a₁ + b₁, a₂ + b₂⟩

• a - b = ⟨a₁ - b₁, a₂ - b₂⟩

• ka = ⟨ka₁, ka₂⟩

Vector Sum

• ai + bj

  • Component Form: ⟨a, b⟩

Direction of Vectors

• |v|(cos θ)i + |v|(sin θ)j

  • a = |v|(cos θ)

  • b = |v|(sin θ)

• tan⁻¹(b/a)

Dot Product

• a ⋅ b = a₁b₁ + a₂b₂

  • Orthogonal when a ⋅ b = 0 ~ except zero vector

Commutative Property of the Dot Product

• u ⋅ v = v ⋅ u

Distributive Property of the Dot Product

• u(v + w) = (u ⋅ v )+ (u ⋅ w)

Scalar Multiplication Property of the Dot Product

• k(uv) = ku ⋅ v = u ⋅ kv

Zero Vector Dot Product Property

• 0 ⋅ u = 0

Dot Product and Vector Magnitude Relationship

• u ⋅ u = |u|²

  • u₁² + u₂²

Angle Between 2 Vectors

• cos θ = (a ⋅ b)/(|a| |b|)

Vector Projection

• projᵥu = [(u ⋅ v)/|v|²]v

  • Projection of u onto v

  • Parallel to v ~ not always the same direction

Work

• W₁ = |F| |AB|

  • |F| (cos θ) |AB|

  • |proj__AB__F| |AB|

    • The first AB should be in subscript

Unit Vector

• (Component Form)/(Magnitude)