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Dot Product from Magnitudes and Included Angle

a⃗⋅b⃗=∣a⃗∣ ∣b⃗∣cos⁡θ\vec{a}\cdot\vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta

Worked example: |a| = 6, |b| = 5, 60° apart → dot = 15 — press Try an example to run it live, then adjust anything.

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Dot Product from Magnitudes and Included Angle explained

θ|a||b|a·b

This is the dot product seen from the geometry side rather than the component side: length times length times the cosine of the angle between. The two views are the same number, which is exactly why the dot product is useful — you can compute it cheaply from coordinates and then read it as a statement about angles. With |a| = 6, |b| = 5 and 60° between them, a·b = 6 × 5 × 0.5 = 15. Push the angle to 90° and the product collapses to zero; push past it and the product turns negative.

The everyday consequence is physical: work is a dot product, so a porter carrying a suitcase horizontally does no work against gravity at all, the force being straight up and the motion straight along. The trap is assuming that θ is any angle in your diagram; it must be the angle between the two vectors when they are placed tail to tail. Nose-to-tail sketches quietly hand you the supplement, and cos flips sign the moment you cross 90°.

Dot Product from Magnitudes and Included Angle formula

a⃗⋅b⃗=∣a⃗∣ ∣b⃗∣cos⁡θ\vec{a}\cdot\vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta
Where
  • a⃗⋅b⃗\vec{a}\cdot\vec{b}= Dot product
  • ∣a⃗∣|\vec{a}|= Magnitude of a
  • ∣b⃗∣|\vec{b}|= Magnitude of b
  • θ\theta= Angle between a and b (°)

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Part of this formula set

Vector operations

12 formulasMagnitude, direction, components and unit vectors, plus the dot and cross products and the projections and angles they give you.