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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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