A worked example: (1,2,3) · (4,5,6)
The dot product of (1,2,3) and (4,5,6) is (1×4) + (2×5) + (3×6) = 4 + 10 + 18 = 32, and the angle between them works out to 12.9332°.
The dot product formula
For two vectors of equal length, the dot product a·b is the sum of each pair of matching components multiplied together: a₁b₁ + a₂b₂ + a₃b₃ + ... It collapses two vectors into a single number, unlike vector addition, which produces another vector.
From dot product to angle: cos θ = (a·b) ÷ (|a||b|)
Rearranging the dot product's geometric definition (a·b = |a||b|cos θ) gives the angle directly: θ = arccos((a·b) ÷ (|a||b|)). For (1,2,3) and (4,5,6), the magnitudes are √14 ≈ 3.7417 and √77 ≈ 8.775, so cos θ = 32 ÷ (3.7417 × 8.775) ≈ 0.9746, and arccos(0.9746) ≈ 12.9332°.
What a zero, positive, or negative dot product means
The sign and size of the dot product read directly as a relationship between the vectors. Two perpendicular vectors like (1,0,0) and (0,1,0) have a dot product of exactly 0 and an angle of exactly 90° — the dot product is zero precisely when vectors are perpendicular. Two opposite-pointing vectors like (2,0,0) and (−3,0,0) give a dot product of −6 and an angle of 180° — a negative dot product means the vectors point in broadly opposite directions.
When to reach for the cross product instead
The dot product works in any dimension and answers "how aligned are these two vectors," as a single scalar number. When the question is instead "what vector is perpendicular to both of these" (only meaningful in 3D), that's the cross product's job — a genuinely different calculation, not just a variant of this one.