Ch. 8 Vectors - Precalculus

Same/opposite direction

Not necessarily same magnitude

Same magnitude

Same direction

Same magnitude

Opposite direction (-a)

The sum of 2 or more vectors

AKA null vector

Magnitude of 0

No specific direction

Resultant of 2 opposite vectors

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

⟨Δx, Δy⟩

• ⟨x₂ - x₁ y₂ - y₁⟩

a + b = ⟨a₁ + b₁, a₂ + b₂⟩

a - b = ⟨a₁ - b₁, a₂ - b₂⟩

ka = ⟨ka₁, ka₂⟩

ai + bj

• Component Form: ⟨a, b⟩

|v|(cos θ)i + |v|(sin θ)j

• a = |v|(cos θ)

• b = |v|(sin θ)

tan⁻¹(b/a)

a ⋅ b = a₁b₁ + a₂b₂

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

u ⋅ v = v ⋅ u

u(v + w) = (u ⋅ v )+ (u ⋅ w)

k(uv) = ku ⋅ v = u ⋅ kv

0 ⋅ u = 0

u ⋅ u = |u|²

• u₁² + u₂²

cos θ = (a ⋅ b)/(|a| |b|)

projᵥu = [(u ⋅ v)/|v|²]v

• Projection of u onto v

• Parallel to v ~ not always the same direction

W₁ = |F| |AB|

• |F| (cos θ) |AB|

• |proj__AB__F| |AB|

  • The first AB should be in subscript

(Component Form)/(Magnitude)