* Same/opposite direction * Not necessarily same magnitude
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Equivalent vectors
* Same magnitude * Same direction
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Opposite vectors
* Same magnitude * Opposite direction (-a)
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Resultant vector
* The sum of 2 or more vectors
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Zero vector
* AKA null vector * Magnitude of 0 * No specific direction * Resultant of 2 opposite vectors
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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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Component Form
* ⟨Δx, Δy⟩ * ⟨x₂ - x₁ y₂ - y₁⟩
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Vector Operations
* a + b = ⟨a₁ + b₁, a₂ + b₂⟩ * a - b = ⟨a₁ - b₁, a₂ - b₂⟩ * ka = ⟨ka₁, ka₂⟩
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Vector Sum
* ai + bj * Component Form: ⟨a, b⟩
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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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Dot Product
* a ⋅ b = a₁b₁ + a₂b₂ * Orthogonal when a ⋅ b = 0 \~ except zero vector
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Commutative Property of the Dot Product
* u ⋅ v = v ⋅ u
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Distributive Property of the Dot Product
* u(v + w) = (u ⋅ v )+ (u ⋅ w)
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Scalar Multiplication Property of the Dot Product
* k(uv) = ku ⋅ v = u ⋅ kv
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Zero Vector Dot Product Property
* 0 ⋅ u = 0
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Dot Product and Vector Magnitude Relationship
* u ⋅ u = |u|² * u₁² + u₂²
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Angle Between 2 Vectors
* cos θ = (a ⋅ b)/(|a| |b|)
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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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Work
* W₁ = |F| |**AB**| * |F| (cos θ) |**AB**| * |proj__*AB*__F| |**AB**| * The first AB should be in subscript