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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
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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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Where Dot Product from Magnitudes and Included Angle fits

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Scalar Projection of One Vector onto Another
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See also

Cross Product Magnitude: Both use vector magnitude and angle between vectors.Resultant of Two Vectors at an Angle: Both use vector magnitude and angle between vectors.Scalar Projection of One Vector onto Another: Both use dot product and vector magnitude.Angle Between Two 2D Vectors (Components): Both use angle between vectors.Dot Product of Two 2D Vectors (Components): Both use dot product.Dot Product of Two 3D Vectors (Components): Both use dot product.
6 more relatedMagnitude of a 2D Vector: Both use vector magnitude.Magnitude of a 3D Vector: Both use vector magnitude.Magnitude of a Scaled Vector: Both use vector magnitude.Unit Vector Component (Normalization): Both use vector magnitude.x-Component from Magnitude and Angle: Both use vector magnitude.y-Component from Magnitude and Angle: Both use vector magnitude.
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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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Vector operations

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

Revised 31 October 2026
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formula.expert. (2026, October 31). Dot Product from Magnitudes and Included Angle Calculator. https://formula.expert/solvers/dot-product-magnitude-angle

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