lightbulb Key Concepts

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1. Definition

The dot product (scalar product) of vectors a and b is:

a · b = |a| |b| cos θ

2. Properties

• a · b = b · a (Commutative)
• a · (b + c) = a · b + a · c (Distributive)
• If a · b = 0, the vectors are perpendicular.
• a · a = |a|²

3. Coordinate Form

If a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, then:

a · b = a₁b₁ + a₂b₂ + a₃b₃