Defining and Computing Dot Product

Trigonometry · Axiom Academy

One operation, two powerful formulas: component-wise multiplication and geometric projection Given two vectors u and v , the dot product (written u · v ) combines them into a single scalar value. This scalar tells us something important about how the vectors relate to each other. The first way to compute the dot product is by multiplying corresponding components and adding them up: For example, if u = ⟨3, 4⟩ and v = ⟨2, 1⟩, then: u · v = (3)(2) + (4)(1) = 6 + 4 = 10 The second way involves the angle θ between the vectors and their magnitudes: Where || u || is the magnitude (length) of u , || v || is the magnitude of v , and θ is the angle between them. Remarkably, both formulas always give the same result! Let's verify with our example u = ⟨3, 4⟩ and v = ⟨2, 1⟩: This equivalence is not a coincidence—it's a fundamental theorem that connects algebra with geometry! The dot product tells us about the projection of one vector onto another. Geometrically, u · v measures "how much of u goes in the direction of v " (or vice versa), scaled by the magnitudes. Positive dot product: Vectors point in generally the same direction (θ < 90°) Zero dot product: Vectors are perpendicular (θ = 90°) Negative dot product: Vectors point in generally opposite directions (θ > 90°) Maximum value: When vectors are parallel (θ = 0°)

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