# Ch. 8 Vectors - Precalculus

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Parallel Vectors

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## Tags and Description

### 20 Terms

1

Parallel Vectors

• Same/opposite direction

• Not necessarily same magnitude

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2

Equivalent vectors

• Same magnitude

• Same direction

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3

Opposite vectors

• Same magnitude

• Opposite direction (-a)

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4

Resultant vector

• The sum of 2 or more vectors

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5

Zero vector

• AKA null vector

• Magnitude of 0

• No specific direction

• Resultant of 2 opposite vectors

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6

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

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7

Component Form

• ⟨Δx, Δy⟩

• ⟨x₂ - x₁ y₂ - y₁⟩

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8

Vector Operations

• a + b = ⟨a₁ + b₁, a₂ + b₂⟩

• a - b = ⟨a₁ - b₁, a₂ - b₂⟩

• ka = ⟨ka₁, ka₂⟩

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9

Vector Sum

• ai + bj

• Component Form: ⟨a, b⟩

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10

Direction of Vectors

• |v|(cos θ)i + |v|(sin θ)j

• a = |v|(cos θ)

• b = |v|(sin θ)

• tan⁻¹(b/a)

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11

Dot Product

• a ⋅ b = a₁b₁ + a₂b₂

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

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12

Commutative Property of the Dot Product

• u ⋅ v = v ⋅ u

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13

Distributive Property of the Dot Product

• u(v + w) = (u ⋅ v )+ (u ⋅ w)

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14

Scalar Multiplication Property of the Dot Product

• k(uv) = ku ⋅ v = u ⋅ kv

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15

Zero Vector Dot Product Property

• 0 ⋅ u = 0

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16

Dot Product and Vector Magnitude Relationship

• u ⋅ u = |u|²

• u₁² + u₂²

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17

Angle Between 2 Vectors

• cos θ = (a ⋅ b)/(|a| |b|)

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18

Vector Projection

• projᵥu = [(u ⋅ v)/|v|²]v

• Projection of u onto v

• Parallel to v ~ not always the same direction

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19

Work

• W₁ = |F| |AB|

• |F| (cos θ) |AB|

• |proj__AB__F| |AB|

• The first AB should be in subscript

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20

Unit Vector

• (Component Form)/(Magnitude)

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